Square Root Questions with Solutions for Exams

Explanation:
To find the square root of 196, we determine which number multiplied by itself results in 196.
Testing the given options, we find that 14 \times 14 = 196.
Therefore, the square root of 196 is 14.

Explanation:
To find the square root of 576, we look for a number which, when multiplied by itself, equals 576.
Testing the given options, we find that 24 \times 24 = 576.
Therefore, the square root of 576 is 24.

Explanation:
To find the square root of 1296, we can test the options by multiplication or use prime factorization.
Squaring option C gives 36 \times 36 = 1296.
Alternatively, using prime factorization, 1296 = 2^4 \times 3^4, and taking the square root yields 2^2 \times 3^2 = 4 \times 9 = 36.
Therefore, the square root of 1296 is 36.

Explanation:
To find the square root of the decimal number 17.64, we can convert it into a fraction to simplify the calculation.
Expressing 17.64 as a fraction yields \frac{1764}{100}.
Taking the square root on both numerator and denominator gives \frac{\sqrt{1764}}{\sqrt{100}}.
Since \sqrt{1764} = 42 and \sqrt{100} = 10, the expression becomes \frac{42}{10} = 4.2.
Alternatively, squaring 4.2 confirms that 4.2 \times 4.2 = 17.64.
Therefore, the correct answer is 4.2.

Explanation:
To find the least number by which 3920 must be multiplied to form a perfect square, we first determine its prime factorization.
Expressing 3920 in terms of prime factors yields 3920 = 2^4 \times 5^1 \times 7^2.
For a number to be a perfect square, all the exponents in its prime factorization must be even integers.
In this factorization, the exponents of 2 and 7 are 4 and 2 (both even), but the exponent of 5 is 1 (odd).
To make the exponent of 5 even, we must multiply by 5 so that its power becomes 5^2.
Therefore, the least number by which 3920 must be multiplied is 5.

Explanation:
To find the least number by which 2940 must be divided to yield a perfect square, we first find its prime factorization.
Expressing 2940 in terms of prime factors gives 2940 = 2^2 \times 3^1 \times 5^1 \times 7^2.
For a number to be a perfect square, all the exponents in its prime factorization must be even integers.
Here, the exponents of 2 and 7 are even, while the exponents of 3 and 5 are 1 (which are odd).
To eliminate these odd powers by division, we must divide the number by 3^1 \times 5^1 = 15.
Therefore, the least number by which 2940 must be divided is 15.

Explanation:
To find the value of the expression, we evaluate the nested square roots starting from the innermost radical.
First, find the square root of the innermost term: \sqrt{169} = 13.
Next, add this value to 51: 51 + 13 = 64.
Then, find the square root of 64: \sqrt{64} = 8.
Add this result to 248: 248 + 8 = 256.
Finally, find the square root of 256: \sqrt{256} = 16.
Therefore, the correct answer is 16.

Explanation:
To find the value of x, we start with the given equation:
\sqrt{x} + \sqrt{x + 16} = 8
Rearrange the terms to isolate one of the square root expressions:
\sqrt{x + 16} = 8 - \sqrt{x}
Square both sides of the equation to eliminate the radical:
(\sqrt{x + 16})^2 = (8 - \sqrt{x})^2
x + 16 = 64 - 16\sqrt{x} + x
Subtract x from both sides of the equation:
16 = 64 - 16\sqrt{x}
Rearrange the terms to solve for the remaining radical term:
16\sqrt{x} = 64 - 16
16\sqrt{x} = 48
Divide both sides by 16:
\sqrt{x} = 3
Squaring both sides gives:
x = 3^2 = 9
Therefore, the value of x is 9.

Explanation:
To find the value of the expression \sqrt{7 + 4\sqrt{3}}, we can express the expression inside the square root as a complete square of the form (a + b)^2 = a^2 + b^2 + 2ab.
We can rewrite 4\sqrt{3} as 2 \times 2 \times \sqrt{3}.
Thus, we split 7 into 4 + 3, giving 4 + 3 + 2(2)(\sqrt{3}).
Expressing 4 and 3 as squares yields 2^2 + (\sqrt{3})^2 + 2(2)(\sqrt{3}).
This matches the algebraic identity (2 + \sqrt{3})^2.
Taking the square root of (2 + \sqrt{3})^2 results in 2 + \sqrt{3}.
Therefore, the correct answer is 2 + \sqrt{3}.

Explanation:
Let the given infinite expression be equal to x.
x = \sqrt{12 + \sqrt{12 + \sqrt{12 + \dots \infty}}}
Squaring both sides of the equation yields:
x^2 = 12 + \sqrt{12 + \sqrt{12 + \dots \infty}}
Since the inner radical expression is identical to x, we can substitute x into the equation:
x^2 = 12 + x
Rearranging the terms into a standard quadratic form gives:
x^2 - x - 12 = 0
Factoring the quadratic equation results in:
(x - 4)(x + 3) = 0
This gives two possible solutions: x = 4 or x = -3.
Since the principal square root of a real number cannot be negative, we discard x = -3.
Therefore, the value of the expression is 4.

Explanation:
To evaluate an expression where a radical is repeated finitely under multiplication, we can use the standard exponent rule.
For an expression of the form \sqrt{a \sqrt{a \dots \text{ (n times)}}}, the equivalent exponent form is given by the formula:
a^{\frac{2^n - 1}{2^n}}
In this given problem, the base a = 7 and the number of radical signs n = 4.
Substituting these values into the formula gives:
7^{\frac{2^4 - 1}{2^4}} = 7^{\frac{16 - 1}{16}} = 7^{\frac{15}{16}}
Therefore, the correct answer is 7^{\frac{15}{16}}.

Explanation:
To evaluate an infinite product of radicals of the form \sqrt{a \sqrt{a \sqrt{a \dots \infty}}}, we can let the entire expression equal x.
x = \sqrt{a \sqrt{a \sqrt{a \dots \infty}}}
Squaring both sides of the equation gives:
x^2 = a \sqrt{a \sqrt{a \sqrt{a \dots \infty}}}
Since the inner expression is equal to x, we substitute x back into the equation:
x^2 = a \cdot x
Assuming x \neq 0, we divide both sides by x:
x = a
For this specific problem, a = 5, so the value of the infinite radical expression is simply 5.
Therefore, the correct answer is 5.

Explanation:
Let the given infinite subtraction radical expression be equal to x.
x = \sqrt{20 - \sqrt{20 - \sqrt{20 - \dots \infty}}}
Squaring both sides of the equation yields:
x^2 = 20 - \sqrt{20 - \sqrt{20 - \dots \infty}}
Since the inner radical expression is identical to x, we substitute x back into the equation:
x^2 = 20 - x
Rearranging the terms into a standard quadratic form gives:
x^2 + x - 20 = 0
Factoring the quadratic equation results in:
(x + 5)(x - 4) = 0
This gives two possible solutions: x = -5 or x = 4.
Since the principal square root of a real number cannot be negative, we discard x = -5.
Therefore, the value of the expression is 4.

Explanation:
To find the value of the given expression, we evaluate each square root term individually.
First, the square root of 0.01 is \sqrt{0.01} = 0.1.
Second, the square root of 0.81 is \sqrt{0.81} = 0.9.
Third, the square root of 1.21 is \sqrt{1.21} = 1.1.
Fourth, the square root of 0.0009 is \sqrt{0.0009} = 0.03.
Adding all these evaluated values together: 0.1 + 0.9 + 1.1 + 0.03 = 2.13.
Therefore, the correct answer is 2.13.

Explanation:
To evaluate the given series, we rationalize each denominator individually by multiplying its numerator and denominator by its conjugate.
For any general term of the form \frac{1}{\sqrt{n} + \sqrt{n+1}}, rationalizing yields \sqrt{n+1} - \sqrt{n}.
Applying this property to each fraction in the expression transforms the series into a telescoping form:
(\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + (\sqrt{5} - \sqrt{4}) + (\sqrt{6} - \sqrt{5}) + (\sqrt{7} - \sqrt{6}) + (\sqrt{8} - \sqrt{7}) + (3 - \sqrt{8})
All intermediate radical terms cancel each other out, leaving only the highest square root and the initial negative term:
\sqrt{9} - 1 = 3 - 1 = 2
Therefore, the correct answer is 2.

Explanation:
To find the least number that must be subtracted from 5607 to make it a perfect square, we can use the long division method to find its square root.
Grouping the digits in pairs from right to left gives 56 and 07.
The greatest number whose square is less than or equal to 56 is 7, since 7 \times 7 = 49.
Subtracting 49 from 56 leaves a remainder of 7, and bringing down the next pair 07 gives 707.
Doubling the current divisor 7 gives 14. We look for a digit n such that 14n \times n \le 707, which yields n = 4 because 144 \times 4 = 576.
Subtracting 576 from 707 leaves a remainder of 131.
This remainder indicates that 5607 exceeds the square of 74 (74^2 = 5476) by 131.
Therefore, subtracting 131 from 5607 will result in the perfect square 5476.

Explanation:
To find the value of x, we start with the given equation:
\sqrt{1 + \frac{x}{144}} = \frac{13}{12}
Square both sides of the equation to eliminate the square root:
\left(\sqrt{1 + \frac{x}{144}}\right)^2 = \left(\frac{13}{12}\right)^2
1 + \frac{x}{144} = \frac{169}{144}
Subtract 1 from both sides of the equation:
\frac{x}{144} = \frac{169}{144} - 1
\frac{x}{144} = \frac{169 - 144}{144} = \frac{25}{144}
Multiply both sides by 144 to solve for x:
x = 25
Therefore, the value of x is 25.

Explanation:
To find the value of x + \frac{1}{x}, we first determine the reciprocal \frac{1}{x}.
Since x = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}, its reciprocal is \frac{1}{x} = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}.
Now, we add x and \frac{1}{x} together:
x + \frac{1}{x} = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}} + \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}
Taking the common denominator, we get:
\frac{(\sqrt{7} - \sqrt{3})^2 + (\sqrt{7} + \sqrt{3})^2}{(\sqrt{7})^2 - (\sqrt{3})^2}
Expanding the numerators using algebraic identities gives:
\frac{(7 - 2\sqrt{21} + 3) + (7 + 2\sqrt{21} + 3)}{7 - 3} = \frac{10 + 10}{4} = \frac{20}{4} = 5
Therefore, the correct answer is 5.

Explanation:
To find the value of the given expression, we can rationalize the denominator to simplify the calculation.
Multiplying both the numerator and the denominator by the conjugate of \sqrt{2} - 1 (which is \sqrt{2} + 1) gives:
\frac{(\sqrt{2} + 1)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} = \frac{(\sqrt{2} + 1)^2}{(\sqrt{2})^2 - 1^2}
Expanding the numerator using algebraic identities and simplifying the denominator results in:
\frac{2 + 1 + 2\sqrt{2}}{2 - 1} = \frac{3 + 2\sqrt{2}}{1} = 3 + 2\sqrt{2}
Now, substitute the given approximate value of \sqrt{2} = 1.4142 into the expression:
3 + 2(1.4142) = 3 + 2.8284 = 5.8284
Therefore, the correct answer is 5.8284.

Explanation:
To find the square root of the decimal number 0.00001225, we can first consider the whole number ignoring the decimal places, which is 1225.
The square root of 1225 is 35, since 35 \times 35 = 1225.
The given number has 8 decimal places. When we take the square root of a decimal number, the number of decimal places is halved, resulting in 8 \div 2 = 4 decimal places.
Adjusting the position of the decimal point for 35 to have 4 decimal places gives 0.0035.
Squaring 0.0035 confirms that 0.0035 \times 0.0035 = 0.00001225.
Therefore, the correct answer is 0.0035.

Explanation:
To find the square root of the decimal number 0.00001225, we can first consider the whole number ignoring the decimal places, which is 1225.
The square root of 1225 is 35, since 35 \times 35 = 1225.
The given number has 8 decimal places. When we take the square root of a decimal number, the number of decimal places is halved, resulting in 8 \div 2 = 4 decimal places.
Adjusting the position of the decimal point for 35 to have 4 decimal places gives 0.0035.
Squaring 0.0035 confirms that 0.0035 \times 0.0035 = 0.00001225.
Therefore, the correct answer is 0.0035.

Explanation:
To find the value of \sqrt{x} - \frac{1}{\sqrt{x}}, we first simplify \sqrt{x} from the given expression for x.
Given x = 3 + 2\sqrt{2}, we can rewrite 3 as 2 + 1 and 2\sqrt{2} as 2(\sqrt{2})(1).
This transforms the expression into (\sqrt{2})^2 + 1^2 + 2(\sqrt{2})(1) = (\sqrt{2} + 1)^2.
Taking the square root gives \sqrt{x} = \sqrt{(\sqrt{2} + 1)^2} = \sqrt{2} + 1.

Next, we find the reciprocal \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{2} + 1}.
Rationalizing the denominator yields \sqrt{2} - 1.
Subtracting the reciprocal from \sqrt{x} gives (\sqrt{2} + 1) - (\sqrt{2} - 1) = \sqrt{2} + 1 - \sqrt{2} + 1 = 2.
Therefore, the correct answer is 2.

Explanation:
To find the value of x, we start with the given radical equation:
\sqrt{x + 3} + \sqrt{x - 2} = 5
Rearrange the equation to isolate one of the square root terms:
\sqrt{x + 3} = 5 - \sqrt{x - 2}
Square both sides of the equation to eliminate the radical:
(\sqrt{x + 3})^2 = (5 - \sqrt{x - 2})^2
x + 3 = 25 - 10\sqrt{x - 2} + (x - 2)
Simplify the right side of the equation:
x + 3 = 23 + x - 10\sqrt{x - 2}

Subtract x from both sides:
3 = 23 - 10\sqrt{x - 2}
Rearrange the terms to isolate the remaining radical term:
10\sqrt{x - 2} = 23 - 3
10\sqrt{x - 2} = 20

Divide both sides by 10:
\sqrt{x - 2} = 2
Square both sides again to solve for x:
x - 2 = 2^2 = 4
x = 4 + 2 = 6
Therefore, the value of x is 6.

Explanation:
To evaluate this nested radical expression, we work from the innermost square root outward:
First, evaluate the innermost square root: \sqrt{225} = 15.
Add this result to 154: 154 + 15 = 169, and its square root is \sqrt{169} = 13.
Add this result to 108: 108 + 13 = 121, and its square root is \sqrt{121} = 11.
Add this result to 25: 25 + 11 = 36, and its square root is \sqrt{36} = 6.
Finally, add this result to 10: 10 + 6 = 16, and take the outer square root: \sqrt{16} = 4.
Therefore, the correct answer is 4.

Explanation:
To find the value of x^2 + \frac{1}{x^2}, we first determine x^2 from the given value of x.
Given x = \sqrt{3} + \sqrt{2}, squaring both sides gives:
x^2 = (\sqrt{3} + \sqrt{2})^2 = (\sqrt{3})^2 + (\sqrt{2})^2 + 2(\sqrt{3})(\sqrt{2}) = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6}
Next, we find the reciprocal \frac{1}{x^2} = \frac{1}{5 + 2\sqrt{6}}.
Rationalizing the denominator by multiplying the numerator and denominator by the conjugate 5 - 2\sqrt{6} yields:
\frac{5 - 2\sqrt{6}}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})} = \frac{5 - 2\sqrt{6}}{5^2 - (2\sqrt{6})^2} = \frac{5 - 2\sqrt{6}}{25 - 24} = 5 - 2\sqrt{6}
Now, adding x^2 and \frac{1}{x^2} together:
(5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10
Therefore, the correct answer is 10.

Explanation:
Let the given infinite expression be equal to x.
x = \sqrt{56 + \sqrt{56 + \sqrt{56 + \dots \infty}}}
Squaring both sides of the equation yields:
x^2 = 56 + \sqrt{56 + \sqrt{56 + \dots \infty}}
Since the inner radical expression is identical to x, we can substitute x into the equation:
x^2 = 56 + x
Rearranging the terms into a standard quadratic form gives:
x^2 - x - 56 = 0
Factoring the quadratic equation results in:
(x - 8)(x + 7) = 0
This gives two possible solutions: x = 8 or x = -7.
Since the principal square root of a real number cannot be negative, we discard x = -7.
Therefore, the value of the expression is 8.

Explanation:
Let the given infinite subtraction radical expression be equal to x.
x = \sqrt{72 - \sqrt{72 - \sqrt{72 - \dots \infty}}}
Squaring both sides of the equation yields:
x^2 = 72 - \sqrt{72 - \sqrt{72 - \dots \infty}}
Since the inner radical expression is identical to x, we substitute x back into the equation:
x^2 = 72 - x

Rearranging the terms into a standard quadratic form gives:
x^2 + x - 72 = 0
Factoring the quadratic equation results in:
(x + 9)(x - 8) = 0
This gives two possible solutions: x = -9 or x = 8.
Since the principal square root of a real number cannot be negative, we discard x = -9.
Alternatively, for expressions of the form \sqrt{N - \sqrt{N - \dots}} where N can be factored into consecutive integers n \times (n+1), the value is always the smaller integer, which is 8 (since 72 = 8 \times 9).
Therefore, the value of the expression is 8.

Explanation:
First, we find the value of x + \frac{1}{x}.
Given x = 2 + \sqrt{3}, its reciprocal is \frac{1}{x} = \frac{1}{2 + \sqrt{3}} = 2 - \sqrt{3}.
Adding them together: x + \frac{1}{x} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4.
Using the algebraic identity for the sum of cubes: x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right).
Substituting the value 4 into the identity:
4^3 - 3(4) = 64 - 12 = 52
Therefore, the correct answer is 52.

Explanation:
To evaluate this nested radical expression, we work from the innermost square root outward:
First, evaluate the innermost square root: \sqrt{49} = 7.
Add this result to 29: 29 + 7 = 36, and its square root is \sqrt{36} = 6.
Add this result to 19: 19 + 6 = 25, and its square root is \sqrt{25} = 5.
Add this result to 11: 11 + 5 = 16, and its square root is \sqrt{16} = 4.
Finally, add this result to 5: 5 + 4 = 9, and take the outer square root: \sqrt{9} = 3.
Therefore, the correct answer is 3.

Explanation:
First, we find the product of x and y:
xy = \left(\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}\right)\left(\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\right) = 1
Next, we find the sum of x and y:
x + y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} + \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{(\sqrt{3} - \sqrt{2})^2 + (\sqrt{3} + \sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2}
Expanding the numerators and simplifying the denominator gives:
x + y = \frac{(3 - 2\sqrt{6} + 2) + (3 + 2\sqrt{6} + 2)}{3 - 2} = \frac{10}{1} = 10
Now, we calculate x^2 + y^2 using the identity x^2 + y^2 = (x + y)^2 - 2xy:
x^2 + y^2 = 10^2 - 2(1) = 100 - 2 = 98
Finally, adding xy to x^2 + y^2 yields:
x^2 + y^2 + xy = 98 + 1 = 99
Therefore, the correct answer is 99.

Explanation:
To find the value of x, we start with the given radical equation:
\sqrt{5x - 6} + \sqrt{5x + 6} = 6
Isolate one of the radical terms:
\sqrt{5x - 6} = 6 - \sqrt{5x + 6}
Square both sides of the equation to eliminate the radical:
(\sqrt{5x - 6})^2 = (6 - \sqrt{5x + 6})^2
5x - 6 = 36 - 12\sqrt{5x + 6} + (5x + 6)
Simplify the right side of the equation:
5x - 6 = 42 + 5x - 12\sqrt{5x + 6}
Subtract 5x from both sides:
-6 = 42 - 12\sqrt{5x + 6}
Isolate the remaining radical term:
12\sqrt{5x + 6} = 42 + 6
12\sqrt{5x + 6} = 48
Divide both sides by 12:
\sqrt{5x + 6} = 4
Square both sides again to solve for x:
5x + 6 = 4^2 = 16
5x = 16 - 6 = 10
x = \frac{10}{5} = 2
Therefore, the correct answer is 2.

Explanation:
First, let’s simplify the expressions for x and y by rationalizing their denominators:
x = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} = \frac{(\sqrt{5} + 1)(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{5 + 1 + 2\sqrt{5}}{5 - 1} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2}
Similarly, for y:
y = \frac{\sqrt{5} - 1}{\sqrt{5} + 1} = \frac{3 - \sqrt{5}}{2}
Next, we find the sum and the product of x and y:
x + y = \frac{3 + \sqrt{5}}{2} + \frac{3 - \sqrt{5}}{2} = \frac{6}{2} = 3
xy = \left(\frac{\sqrt{5} + 1}{\sqrt{5} - 1}\right)\left(\frac{\sqrt{5} - 1}{\sqrt{5} + 1}\right) = 1
Now, using the algebraic identity x^2 + y^2 = (x + y)^2 - 2xy:
x^2 + y^2 = 3^2 - 2(1) = 9 - 2 = 7
Therefore, the correct answer is 7.

Explanation:
First, we simplify the expressions for x and y by writing them as complete squares:
x = \sqrt{7 + 4\sqrt{3}} = \sqrt{4 + 3 + 2(2)(\sqrt{3})} = \sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3}
y = \sqrt{7 - 4\sqrt{3}} = \sqrt{4 + 3 - 2(2)(\sqrt{3})} = \sqrt{(2 - \sqrt{3})^2} = 2 - \sqrt{3}

Next, we find the sum and the product of x and y:
x + y = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4
xy = (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1

To find x^5 + y^5, we can compute lower power sums s_n = x^n + y^n:
1. Find s_2 = x^2 + y^2 = (x + y)^2 - 2xy = 4^2 - 2(1) = 14.
2. Find s_3 = x^3 + y^3 = (x + y)(x^2 - xy + y^2) = 4(14 - 1) = 4(13) = 52.

Now, we use the product of s_2 and s_3 to determine s_5:
s_2 \times s_3 = (x^2 + y^2)(x^3 + y^3) = x^5 + x^2y^3 + x^3y^2 + y^5 = x^5 + y^5 + (xy)^2(x + y)
Substitute the known values into the equation:
14 \times 52 = (x^5 + y^5) + (1)^2(4) 728 = (x^5 + y^5) + 4 x^5 + y^5 = 728 - 4 = 724
Therefore, the correct answer is 724.

Explanation:
To find the value of \frac{x^4 - 1}{x^2}, we can simplify the expression by splitting the fraction:
\frac{x^4 - 1}{x^2} = \frac{x^4}{x^2} - \frac{1}{x^2} = x^2 - \frac{1}{x^2}
Using the difference of squares identity, we can factor this expression into:
x^2 - \frac{1}{x^2} = \left(x + \frac{1}{x}\right)\left(x - \frac{1}{x}\right)
Given x = \sqrt{3} + \sqrt{2}, its reciprocal is:
\frac{1}{x} = \frac{1}{\sqrt{3} + \sqrt{2}} = \sqrt{3} - \sqrt{2}

Next, we find the sum and the difference of x and \frac{1}{x}:
x + \frac{1}{x} = (\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) = 2\sqrt{3}
x - \frac{1}{x} = (\sqrt{3} + \sqrt{2}) - (\sqrt{3} - \sqrt{2}) = 2\sqrt{2}

Now, substitute these values back into the factored expression:
\left(2\sqrt{3}\right) \times \left(2\sqrt{2}\right) = 4\sqrt{6}
Therefore, the correct answer is 4\sqrt{6}.

Explanation:
First, notice that y is the reciprocal of x, meaning their product is:
xy = \left(\frac{\sqrt{5} - 2}{\sqrt{5} + 2}\right)\left(\frac{\sqrt{5} + 2}{\sqrt{5} - 2}\right) = 1
Next, we simplify the expressions for x and y by rationalizing their denominators:
x = \frac{(\sqrt{5} - 2)(\sqrt{5} - 2)}{(\sqrt{5} + 2)(\sqrt{5} - 2)} = \frac{5 + 4 - 4\sqrt{5}}{5 - 4} = 9 - 4\sqrt{5}
y = \frac{(\sqrt{5} + 2)(\sqrt{5} + 2)}{(\sqrt{5} - 2)(\sqrt{5} + 2)} = \frac{5 + 4 + 4\sqrt{5}}{5 - 4} = 9 + 4\sqrt{5}

Now, find the sum of x and y:
x + y = (9 - 4\sqrt{5}) + (9 + 4\sqrt{5}) = 18
Using the algebraic identity for the sum of cubes:
x^3 + y^3 = (x + y)^3 - 3xy(x + y)

Substitute the known values x + y = 18 and xy = 1 into the formula:
x^3 + y^3 = 18^3 - 3(1)(18) = 5832 - 54 = 5778
Therefore, the correct answer is 5778.

Explanation:
To solve this expression, we rationalize each fraction individually by multiplying its numerator and denominator by the conjugate of the denominator:
1. First term:
\frac{1}{3 - \sqrt{8}} = \frac{3 + \sqrt{8}}{(3 - \sqrt{8})(3 + \sqrt{8})} = \frac{3 + \sqrt{8}}{9 - 8} = 3 + \sqrt{8}
2. Second term:
\frac{1}{\sqrt{8} - \sqrt{7}} = \frac{\sqrt{8} + \sqrt{7}}{8 - 7} = \sqrt{8} + \sqrt{7}
3. Third term:
\frac{1}{\sqrt{7} - \sqrt{6}} = \frac{\sqrt{7} + \sqrt{6}}{7 - 6} = \sqrt{7} + \sqrt{6}
4. Fourth term:
\frac{1}{\sqrt{6} - \sqrt{5}} = \frac{\sqrt{6} + \sqrt{5}}{6 - 5} = \sqrt{6} + \sqrt{5}
5. Fifth term:
\frac{1}{\sqrt{5} - 2} = \frac{\sqrt{5} + 2}{5 - 4} = \sqrt{5} + 2 (noting that 2 = \sqrt{4})
Now, substitute these rationalized forms back into the original expression respecting the signs:
(3 + \sqrt{8}) - (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) - (\sqrt{6} + \sqrt{5}) + (\sqrt{5} + 2)
Expanding the parentheses and canceling out the intermediate terms:
= 3 + \sqrt{8} - \sqrt{8} - \sqrt{7} + \sqrt{7} + \sqrt{6} - \sqrt{6} - \sqrt{5} + \sqrt{5} + 2
= 3 + 2 = 5
Therefore, the correct answer is 5.

Explanation:
To find the value of x^4 - 4x^2, we start by squaring both sides of the given equation to simplify it:
x = \sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}
Squaring both sides:
x^2 = \left(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}\right)^2
Using the algebraic identity (a + b)^2 = a^2 + b^2 + 2ab:
x^2 = (\sqrt{5} + 2) + (\sqrt{5} - 2) + 2\sqrt{(\sqrt{5} + 2)(\sqrt{5} - 2)}
Simplifying the terms inside the expression:
x^2 = 2\sqrt{5} + 2\sqrt{(\sqrt{5})^2 - 2^2}
x^2 = 2\sqrt{5} + 2\sqrt{5 - 4}
x^2 = 2\sqrt{5} + 2(1) = 2\sqrt{5} + 2
Next, rearrange the equation to isolate the radical term:
x^2 - 2 = 2\sqrt{5}
Square both sides again to eliminate the square root:
(x^2 - 2)^2 = (2\sqrt{5})^2
x^4 - 4x^2 + 4 = 20
Subtracting 4 from both sides yields:
x^4 - 4x^2 = 16
Therefore, the correct answer is 16.

Explanation:
To find the value of x^4 - 4x^2, we start by squaring both sides of the given equation to simplify it:
x = \sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}
Squaring both sides:
x^2 = \left(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}\right)^2
Using the algebraic identity (a + b)^2 = a^2 + b^2 + 2ab:
x^2 = (\sqrt{5} + 2) + (\sqrt{5} - 2) + 2\sqrt{(\sqrt{5} + 2)(\sqrt{5} - 2)}
Simplifying the terms inside the expression:
x^2 = 2\sqrt{5} + 2\sqrt{(\sqrt{5})^2 - 2^2}
x^2 = 2\sqrt{5} + 2\sqrt{5 - 4}
x^2 = 2\sqrt{5} + 2(1) = 2\sqrt{5} + 2
Next, rearrange the equation to isolate the radical term:
x^2 - 2 = 2\sqrt{5}
Square both sides again to eliminate the square root:
(x^2 - 2)^2 = (2\sqrt{5})^2
x^4 - 4x^2 + 4 = 20
Subtracting 4 from both sides yields:
x^4 - 4x^2 = 16
Therefore, the correct answer is 16.

Explanation:
To find the value of the expression without having to solve for x directly, we can use algebraic properties of conjugates and differences of squares.
Let A = \sqrt{x^2 + x + 1} and B = \sqrt{x^2 + x - 7}.
From the problem statement, we are given:
A - B = 2
Now, let’s look at the difference of the squares of A and B:
A^2 - B^2 = (x^2 + x + 1) - (x^2 + x - 7) = 1 - (-7) = 8
Using the algebraic identity for the difference of squares, A^2 - B^2 = (A - B)(A + B), we can substitute our known values:
8 = 2 \times (A + B)
Dividing both sides by 2 gives:
A + B = \frac{8}{2} = 4
Therefore, the value of the expression \sqrt{x^2 + x + 1} + \sqrt{x^2 + x - 7} is 4, and the correct option is B.

Explanation:
To evaluate the expression, we can divide each term in the numerator by the denominator x^3:
\frac{x^6 + x^4 + x^2 + 1}{x^3} = \frac{x^6}{x^3} + \frac{x^4}{x^3} + \frac{x^2}{x^3} + \frac{1}{x^3} = x^3 + x + \frac{1}{x} + \frac{1}{x^3}
Grouping the terms, we get:
\left(x^3 + \frac{1}{x^3}\right) + \left(x + \frac{1}{x}\right)
Given x = 3 + 2\sqrt{2}, its reciprocal is:
\frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} = 3 - 2\sqrt{2}
Now, find the sum of x and its reciprocal:
x + \frac{1}{x} = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6
Next, use the algebraic identity for the sum of cubes to find x^3 + \frac{1}{x^3}:
x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right)
Substitute x + \frac{1}{x} = 6 into the identity:
x^3 + \frac{1}{x^3} = 6^3 - 3(6) = 216 - 18 = 198
Finally, substitute these values back into our grouped expression:
198 + 6 = 204
Therefore, the correct answer is 204.

Explanation:
First, we find the product of x and y:
xy = \left(\frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}\right)\left(\frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\right) = 1
Next, we find the sum of x and y:
x + y = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}} + \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} = \frac{(\sqrt{7} - \sqrt{3})^2 + (\sqrt{7} + \sqrt{3})^2}{(\sqrt{7})^2 - (\sqrt{3})^2}
Expanding and simplifying the numerator and denominator:
x + y = \frac{(7 - 2\sqrt{21} + 3) + (7 + 2\sqrt{21} + 3)}{7 - 3} = \frac{20}{4} = 5
Now, compute x^2 + y^2 using the algebraic identity x^2 + y^2 = (x + y)^2 - 2xy:
x^2 + y^2 = 5^2 - 2(1) = 25 - 2 = 23
Substitute these values into the target expression 3x^2 + 5xy + 3y^2 = 3(x^2 + y^2) + 5xy:
= 3(23) + 5(1) = 69 + 5 = 74
Therefore, the correct answer is 74.

Explanation:
First, we simplify the expressions for x and y by rationalizing their denominators:
x = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - 1^2} = \frac{3 + 1 + 2\sqrt{3}}{2} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}
y = \frac{(\sqrt{3} - 1)^2}{(\sqrt{3})^2 - 1^2} = \frac{3 + 1 - 2\sqrt{3}}{2} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3}
Next, we use the algebraic identity for the difference of cubes in the numerator:
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
Substitute this identity into the given expression:
\frac{x^3 - y^3}{x^2 + xy + y^2} = \frac{(x - y)(x^2 + xy + y^2)}{x^2 + xy + y^2} = x - y
Now, substitute the simplified values of x and y:
x - y = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2\sqrt{3}
Therefore, the correct answer is 2\sqrt{3} (Option B).

Explanation:
To evaluate the expression using the properties of minimal polynomials, we start by manipulating the given value of x:
x = \frac{\sqrt{5} + 1}{2}
Multiply both sides by 2:
2x = \sqrt{5} + 1
Rearrange to isolate the square root:
2x - 1 = \sqrt{5}
Square both sides of the equation to eliminate the radical:
(2x - 1)^2 = (\sqrt{5})^2
4x^2 - 4x + 1 = 5
4x^2 - 4x - 4 = 0
Dividing the entire equation by 4 gives the characteristic quadratic relation:
x^2 - x - 1 = 0 \implies x^2 = x + 1
Now, we use this relation to express higher powers of x in terms of lower powers:
1. Find x^3:
x^3 = x \cdot x^2 = x(x + 1) = x^2 + x
Substituting x^2 = x + 1 into the expression:
x^3 = (x + 1) + x = 2x + 1
2. Find 2x^2:
2x^2 = 2(x + 1) = 2x + 2
Now, substitute these derived values back into the target expression x^3 - 2x^2 + 1:
(2x + 1) - (2x + 2) + 1 = 2x + 1 - 2x - 2 + 1 = 0
Therefore, the correct answer is 0 (Option A).

Explanation:
First, we simplify the expressions for x and y by rationalizing their denominators:
x = \frac{(\sqrt{7} + \sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{7 + 5 + 2\sqrt{35}}{7 - 5} = \frac{12 + 2\sqrt{35}}{2} = 6 + \sqrt{35}
Similarly, for y:
y = \frac{(\sqrt{7} - \sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{7 + 5 - 2\sqrt{35}}{7 - 5} = \frac{12 - 2\sqrt{35}}{2} = 6 - \sqrt{35}
Next, we find the sum and the product of x and y:
x + y = (6 + \sqrt{35}) + (6 - \sqrt{35}) = 12
xy = (6 + \sqrt{35})(6 - \sqrt{35}) = 6^2 - (\sqrt{35})^2 = 36 - 35 = 1
Now, we calculate x^2 + y^2 using the identity x^2 + y^2 = (x + y)^2 - 2xy:
x^2 + y^2 = 12^2 - 2(1) = 144 - 2 = 142
Substitute these values into the numerator and denominator of the given expression:
Numerator: x^2 - xy + y^2 = (x^2 + y^2) - xy = 142 - 1 = 141
Denominator: x^2 + xy + y^2 = (x^2 + y^2) + xy = 142 + 1 = 143
Thus, the value of the expression is \frac{141}{143}.
Therefore, the correct answer is option D.

Explanation:
First, we find the reciprocal of x:
\frac{1}{x} = \frac{1}{\sqrt{3} + \sqrt{2}} = \sqrt{3} - \sqrt{2}
Next, we calculate the sum of x and its reciprocal:
x + \frac{1}{x} = (\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) = 2\sqrt{3}
To find x^5 + \frac{1}{x^5}, we first compute the lower power sums s_2 = x^2 + \frac{1}{x^2} and s_3 = x^3 + \frac{1}{x^3}:
1. s_2 = \left(x + \frac{1}{x}\right)^2 - 2 = (2\sqrt{3})^2 - 2 = 12 - 2 = 10
2. s_3 = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) = (2\sqrt{3})^3 - 3(2\sqrt{3}) = 24\sqrt{3} - 6\sqrt{3} = 18\sqrt{3}
Now, we use the product of s_2 and s_3 to determine s_5:
s_2 \times s_3 = \left(x^2 + \frac{1}{x^2}\right)\left(x^3 + \frac{1}{x^3}\right) = x^5 + \frac{1}{x^5} + x + \frac{1}{x}
Substitute the known values into the equation:
10 \times 18\sqrt{3} = \left(x^5 + \frac{1}{x^5}\right) + 2\sqrt{3} 180\sqrt{3} = \left(x^5 + \frac{1}{x^5}\right) + 2\sqrt{3} x^5 + \frac{1}{x^5} = 180\sqrt{3} - 2\sqrt{3} = 178\sqrt{3} Therefore, the correct answer is 178\sqrt{3} (Option D).

Explanation:
To evaluate the expression using the properties of minimal polynomials, we start by squaring both sides of the given equation:
x = \sqrt{3} + \sqrt{2}
x^2 = (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6}
Rearrange the equation to isolate the radical term:
x^2 - 5 = 2\sqrt{6}
Square both sides again to eliminate the square root:
(x^2 - 5)^2 = (2\sqrt{6})^2
x^4 - 10x^2 + 25 = 24
Rearranging this gives the characteristic minimal polynomial relation:
x^4 - 10x^2 + 1 = 0
Now, we use this relation to evaluate the target expression x^4 - 10x^2 + 6 by rewriting it as:
x^4 - 10x^2 + 6 = (x^4 - 10x^2 + 1) + 5
Since x^4 - 10x^2 + 1 = 0, we substitute 0 into the expression:
0 + 5 = 5
Therefore, the correct answer is 5 (Option B).

Explanation:
To evaluate the expression, we start by cubing both sides of the given equation:
x = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}
Using the algebraic identity for the cube of a binomial, (a + b)^3 = a^3 + b^3 + 3ab(a + b):
x^3 = (2 + \sqrt{5}) + (2 - \sqrt{5}) + 3\left(\sqrt[3]{(2 + \sqrt{5})(2 - \sqrt{5})}\right)\left(\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}\right)
Simplify the product inside the cube root:
(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
Since the binomial sum \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} is equal to x, we substitute x back into the equation:
x^3 = 4 + 3(-1)(x)
x^3 = 4 - 3x
Rearranging the terms gives the polynomial relation:
x^3 + 3x = 4
Now, substitute this result into the target expression x^3 + 3x + 2:
4 + 2 = 6
Therefore, the correct answer is 6 (Option C).

Explanation:
To evaluate the expression, we start with the given equation:
x = \sqrt[3]{2} + \sqrt[3]{4}
Cube both sides of the equation:
x^3 = (\sqrt[3]{2} + \sqrt[3]{4})^3
Using the algebraic identity for the cube of a binomial, (a + b)^3 = a^3 + b^3 + 3ab(a + b):
x^3 = (\sqrt[3]{2})^3 + (\sqrt[3]{4})^3 + 3(\sqrt[3]{2})(\sqrt[3]{4})(\sqrt[3]{2} + \sqrt[3]{4})
Simplify the individual components:
1. Cubing the terms: (\sqrt[3]{2})^3 = 2 and (\sqrt[3]{4})^3 = 4.
2. Multiplying the radical product: (\sqrt[3]{2})(\sqrt[3]{4}) = \sqrt[3]{8} = 2.
3. Substituting x back for the binomial sum \sqrt[3]{2} + \sqrt[3]{4}:
x^3 = 2 + 4 + 3(2)(x)
x^3 = 6 + 6x
Rearranging the terms to evaluate the target expression x^3 - 6x:
x^3 - 6x = 6
Therefore, the correct answer is 6 (Option B).

Explanation:
To evaluate the expression, we start by cubing both sides of the given equation:
x = \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}}
Using the algebraic identity for the cube of a binomial, (a + b)^3 = a^3 + b^3 + 3ab(a + b):
x^3 = (5 + 2\sqrt{13}) + (5 - 2\sqrt{13}) + 3\left(\sqrt[3]{(5 + 2\sqrt{13})(5 - 2\sqrt{13})}\right)\left(\sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}}\right)
Simplify the product inside the cube root:
(5 + 2\sqrt{13})(5 - 2\sqrt{13}) = 5^2 - (2\sqrt{13})^2 = 25 - 4(13) = 25 - 52 = -27
Taking the cube root gives \sqrt[3]{-27} = -3.
Since the binomial sum \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}} is equal to x, we substitute x back into the equation:
x^3 = 10 + 3(-3)(x)
x^3 = 10 - 9x
Rearranging the terms gives the polynomial relation:
x^3 + 9x = 10
Therefore, the correct answer is 10 (Option B).

Explanation:
To evaluate the expression, we start by cubing both sides of the given equation:
x = \sqrt[3]{9} + \sqrt[3]{3}
Using the algebraic identity for the cube of a binomial, (a + b)^3 = a^3 + b^3 + 3ab(a + b):
x^3 = (\sqrt[3]{9})^3 + (\sqrt[3]{3})^3 + 3(\sqrt[3]{9})(\sqrt[3]{3})(\sqrt[3]{9} + \sqrt[3]{3})
Simplify the individual components:
1. Cubing the terms: (\sqrt[3]{9})^3 = 9 and (\sqrt[3]{3})^3 = 3.
2. Multiplying the radical product: (\sqrt[3]{9})(\sqrt[3]{3}) = \sqrt[3]{27} = 3.
3. Substituting x back for the binomial sum \sqrt[3]{9} + \sqrt[3]{3}:
x^3 = 9 + 3 + 3(3)(x)
x^3 = 12 + 9x
Rearranging the terms gives the polynomial relation:
x^3 - 9x = 12
Now, substitute this result into the target expression x^3 - 9x - 5:
12 - 5 = 7
Therefore, the correct answer is 7 (Option C).