Question 1

$\sqrt{x} + 14 = \sqrt{2601}$

Accepted Answer

Concept & Logic To solve this equation, you must first evaluate the perfect square root on the right side of the equation. Once calculated, use basic algebra to isolate \sqrt{x} and then square both sides to find x. If \sqrt{x} = y, then x = y^2 Step-by-Step Solution • Given: \sqrt{x} + 14 = \sqrt{2601} • Calculation: First, determine the square root of 2601. We know 50^2 = 2500. Since 2601 ends in 1, its square root must end in 1 or 9. The number 2601 is very close to 2500, so the root is 51. \sqrt{2601} = 51 • Substitute this back into the equation: \sqrt{x} + 14 = 51 • Isolate \sqrt{x} by subtracting 14 from both sides: \sqrt{x} = 51 - 14 \sqrt{x} = 37 • Square both sides to solve for x: x = 37^2 x = 1369 Exam Strategy & Shortcut Memorize squares up to 40. Knowing 37^2 = 1369 instantly saves you from having to do manual multiplication during the final step. To quickly verify 37^2, you can use the (a-b)^2 method: (40-3)^2 = 1600 - 240 + 9 = 1369. Common Pitfall A frequent mistake is stopping at \sqrt{x} = 37 and looking for 37 in the options (if it were present), or incorrectly doubling 37 instead of squaring it. Always remember to apply the inverse operation (squaring) to remove the radical. Final Answer Therefore, the correct answer is 1369.

Question 2

If $a = \frac{\sqrt{3}}{2}$, then   $\sqrt{1+a} + \sqrt{1-a} = $ $x$

Accepted Answer

Concept & Formula Instead of plugging in the complex fraction directly under nested radicals, it is much easier to square the entire target expression. This takes advantage of the (x+y)^2 identity to eliminate the outer square roots and simplify the inner terms via the difference of squares. (x+y)^2 = x^2 + y^2 + 2xy (1+a)(1-a) = 1 - a^2 Step-by-Step Solution • Given: a = \frac{\sqrt{3}}{2} and we need to evaluate y = \sqrt{1+a} + \sqrt{1-a} • Calculation: Square the entire expression we want to evaluate: y^2 = (\sqrt{1+a} + \sqrt{1-a})^2 y^2 = (\sqrt{1+a})^2 + (\sqrt{1-a})^2 + 2(\sqrt{1+a})(\sqrt{1-a}) y^2 = (1+a) + (1-a) + 2\sqrt{(1+a)(1-a)} • Simplify the terms: y^2 = 2 + 2\sqrt{1 - a^2} • Now, substitute the given value of a: Since a = \frac{\sqrt{3}}{2}, then a^2 = (3/4) • Plug a^2 into our simplified equation: y^2 = 2 + 2\sqrt{1 - (3/4)} y^2 = 2 + 2\sqrt{(1/4)} y^2 = 2 + 2((1/2)) y^2 = 2 + 1 y^2 = 3 • Solve for y (the original expression): y = \sqrt{3} (Taking the positive root since the sum of positive square roots must be positive). Exam Strategy & Shortcut Whenever you see a pattern like \sqrt{1+a} \pm \sqrt{1-a}, immediately square the expression. It elegantly collapses the a terms and creates a simple 1-a^2 under the remaining radical. This is a standard competitive exam trick that bypasses nested radical denesting algorithms entirely. Common Pitfall Attempting to evaluate \sqrt{1 + \frac{\sqrt{3}}{2}} directly requires finding a perfect square in the form of (x+y)^2 hidden under the root (e.g., multiplying top and bottom by 2 to get \frac{\sqrt{4+2\sqrt{3}}}{\sqrt{2}}). While mathematically possible, it is incredibly time-consuming and prone to arithmetic errors. Final Answer Therefore, the correct answer is \sqrt{3}.

Question 3

What is   $\frac{5+\sqrt{10}}{5\sqrt{5}-2\sqrt{20}-\sqrt{32}+\sqrt{50}}$   equal to

Accepted Answer

Concept & Formula To simplify complex radical fractions, you must break down every surd into its simplest form by extracting perfect squares. Once the denominator is simplified, look for common factors to factor out in both the numerator and denominator to cancel terms. \sqrt{a^2b} = a\sqrt{b} Step-by-Step Solution • Given: \frac{5+\sqrt{10}}{5\sqrt{5}-2\sqrt{20}-\sqrt{32}+\sqrt{50}} • Calculation: First, simplify each term in the denominator independently: 5\sqrt{5} (Already simplified) 2\sqrt{20} = 2\sqrt{4 × 5} = 2(2\sqrt{5}) = 4\sqrt{5} \sqrt{32} = \sqrt{16 × 2} = 4\sqrt{2} \sqrt{50} = \sqrt{25 × 2} = 5\sqrt{2} • Reconstruct and simplify the denominator: = 5\sqrt{5} - 4\sqrt{5} - 4\sqrt{2} + 5\sqrt{2} Group like terms: = (5 - 4)\sqrt{5} + (-4 + 5)\sqrt{2} = \sqrt{5} + \sqrt{2} • The full expression is now: \frac{5+\sqrt{10}}{\sqrt{5}+\sqrt{2}} • Now, look at the numerator. Factor out \sqrt{5}: Notice that 5 = \sqrt{25} = \sqrt{5} × \sqrt{5} and \sqrt{10} = \sqrt{5} × \sqrt{2}. 5 + \sqrt{10} = \sqrt{5}(\sqrt{5} + \sqrt{2}) • Substitute this back into the fraction: = \frac{\sqrt{5}(\sqrt{5} + \sqrt{2})}{\sqrt{5}+\sqrt{2}} • Cancel the common binomial term (\sqrt{5} + \sqrt{2}) from the top and bottom: = \sqrt{5} Exam Strategy & Shortcut Don't immediately jump to rationalizing the denominator. Rationalizing \sqrt{5} + \sqrt{2} by multiplying by its conjugate \sqrt{5} - \sqrt{2} works, but it takes an extra step. In exam settings, numerators are almost always designed to factor cleanly into a multiple of the simplified denominator. Always check for a common factor first! Common Pitfall A common mistake is incorrectly simplifying the surds (e.g., extracting the wrong perfect square) which prevents the denominator from collapsing into a simple binomial. Always double-check your prime factorizations for numbers like 20, 32, and 50. Final Answer Therefore, the correct answer is \sqrt{5}.

Question 4

The square root of   $\frac{(0.75)^3}{1-0.75} + [0.75 + (0.75)^2 + 1]$   is

Accepted Answer

Concept & Formula This problem elegantly masks a standard algebraic identity using decimals. By substituting a variable for the decimal, you can reveal the difference of cubes factorization, which perfectly collapses the entire complex expression. a^3 - b^3 = (a-b)(a^2+ab+b^2) Step-by-Step Solution • Given: We need to find the square root of ((0.75)^3/1-0.75) + [0.75 + (0.75)^2 + 1] • Calculation: Let x = 0.75. Substitute this into the expression: (x^3/1-x) + (x + x^2 + 1) Rearranging the second part slightly: = (x^3/1-x) + (1 + x + x^2) • Combine the terms by finding a common denominator, which is (1-x): = (x^3 + (1-x)(1 + x + x^2)/1-x) • Recognize the algebraic identity in the numerator. The term (1-x)(1 + x + x^2) is exactly the expansion of (1^3 - x^3): = (x^3 + (1 - x^3)/1-x) • The x^3 terms cancel out perfectly: = (1/1-x) • Substitute x = 0.75 back into our simplified expression: = (1/1 - 0.75) = (1/0.25) = (1/\frac{1){4}} = 4 • Finally, the question asks for the square root of this entire expression: \sqrt{4} = 2 Exam Strategy & Shortcut Whenever you see a fraction with a cube, and a polynomial that looks like a^2 + a + 1, immediately think of the difference of cubes identity. Writing out (0.75)^3 mathematically is a waste of time. Replace ugly decimals with a variable (x) to instantly reveal the underlying algebraic structure. Common Pitfall The most dangerous pitfall is successfully simplifying the massive expression down to 4, and then selecting 4 as the answer (which is an option!). You must carefully read the first three words of the question: "The square root of...". You must take the square root of 4 to finish the problem. Final Answer Therefore, the correct answer is 2.

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