Question 1

Simplify the ratio of square roots: What is √288 / √128 equal to (in simplest form)?

Accepted Answer

Introduction / Context: Ratios of square roots can be simplified by combining under one radical and reducing the fraction before taking the square root. Given Data / Assumptions: Expression: √288 / √128. Concept / Approach: Use √a / √b = √(a/b). Reduce 288/128 to lowest terms first, then take the square root to avoid messy radicals. Step-by-Step Solution: √288 / √128 = √(288/128)288/128 = 2.25 = 9/4√(9/4) = 3/2 Verification / Alternative check: Factor approach: 288 = 2^5 * 3^2; 128 = 2^7. Then √288 = 2^2 * 3 * √2 = 12√2; √128 = 2^3 * √2 = 8√2; ratio = (12√2)/(8√2) = 3/2. Why Other Options Are Wrong: √3/2 and √6/2 retain radicals unnecessarily; 3/√2 is not equal to 3/2 here; 2/3 is the reciprocal. Common Pitfalls: Not simplifying the fraction under the radical or canceling √2 incorrectly. Final Answer: 3/2

Question 2

Evaluate the product of roots: What is √10 × √15 equal to?

Accepted Answer

Introduction / Context: Products of square roots are simplified using √a * √b = √(ab). Factor the product to extract perfect squares. Given Data / Assumptions: Compute √10 * √15. Concept / Approach: Multiply under a single radical: √(10 * 15) = √150. Factor 150 as 25 * 6 to extract a perfect square. Step-by-Step Solution: √10 * √15 = √150150 = 25 * 6 ⇒ √150 = √25 * √6 = 5√6 Verification / Alternative check: Approximate: √10 ≈ 3.1623, √15 ≈ 3.8730, product ≈ 12.247; 5√6 ≈ 5 * 2.449 = 12.247. Matches. Why Other Options Are Wrong: 6√5 equals √180, not √150; √30 is too small; 5 is a drastic underestimate; 10√15 is unrelated. Common Pitfalls: Adding under the radical (√10 + √15) instead of multiplying; not extracting the largest perfect square factor. Final Answer: 5√6

Question 3

Solve for the missing number: If √(?) / 200 = 0.02, then what is the value of ?

Accepted Answer

Introduction / Context: This is a direct equation with a square root. Isolate the root term first, then square both sides to remove the radical. Given Data / Assumptions: √(?) / 200 = 0.02. Concept / Approach: Multiply both sides by 200 to isolate √(?), then square. Step-by-Step Solution: √(?) = 0.02 * 200 = 4? = 4^2 = 16 Verification / Alternative check: Plug back: √16 / 200 = 4 / 200 = 0.02. Correct. Why Other Options Are Wrong: 0.4, 1.6, 4, 400 stem from stopping early or squaring/multiplying incorrectly. Common Pitfalls: Squaring only one side or forgetting to isolate the square root before squaring. Final Answer: 16

Question 4

Solve for the missing number: 250 / √(?) = 10. What is the value of ?

Accepted Answer

Introduction / Context: A simple radical equation in the denominator. Clear the denominator first, then isolate the square root, square, and solve. Given Data / Assumptions: 250 / √(?) = 10. Concept / Approach: Cross-multiply to get √(?) in terms of known numbers, then square. Step-by-Step Solution: 250 = 10 * √(?) ⇒ √(?) = 25? = 25^2 = 625 Verification / Alternative check: Check: 250 / √625 = 250 / 25 = 10. Why Other Options Are Wrong: 25 and 250 are values of √(?) or the original numerator; 2500 over-squares; 100 is unrelated. Common Pitfalls: Squaring before isolating the root, leading to algebraic mistakes. Final Answer: 625

Question 5

Evaluate the radical quotient: What is √4375 / √7 equal to?

Accepted Answer

Introduction / Context: When dividing square roots, combine them into a single radical to simplify. Look for perfect-square simplifications. Given Data / Assumptions: Expression: √4375 / √7. Concept / Approach: Use √a / √b = √(a/b). If the fraction simplifies to a perfect square, the square root becomes an integer. Step-by-Step Solution: √4375 / √7 = √(4375 / 7)Since 7 * 625 = 4375, the fraction equals 625√625 = 25 Verification / Alternative check: 4375 = 7 * 625 confirms the simplification; 625 is 25^2. Why Other Options Are Wrong: 24.75 and 27.25 are near-by distractors; 35 and 20 are unrelated integers not supported by the exact simplification. Common Pitfalls: Rationalizing or approximating too early; here, exact simplification is cleaner. Final Answer: 25

Question 6

Compute the difference of radicals: What is √(4/3) − √(3/4) equal to?

Accepted Answer

Introduction / Context: This problem involves subtracting two simple radicals with fractional arguments. Express each in simplified radical form, then subtract using a common denominator. Given Data / Assumptions: Expression: √(4/3) − √(3/4). Concept / Approach: √(4/3) = 2/√3 and √(3/4) = √3/2. Subtract the two by writing over a common denominator. Step-by-Step Solution: √(4/3) = 2/√3√(3/4) = √3/2Difference = 2/√3 − √3/2Common denominator = 2√3 ⇒ (4 − 3) / (2√3) = 1/(2√3) Verification / Alternative check: Approximate numerically: √(4/3) ≈ 1.1547; √(3/4) ≈ 0.8660; difference ≈ 0.2887; 1/(2√3) ≈ 1/(2 * 1.732) ≈ 0.2887. Why Other Options Are Wrong: The negative option reverses order; 1 and 2√3 are far off; 0 implies equality which is false. Common Pitfalls: Subtracting numerators or denominators directly without a common denominator; forgetting to rationalize is fine here since the exact form 1/(2√3) is acceptable. Final Answer: 1/(2√3)

Question 7

Approximation using √2 ≈ 1.4142: Find the approximate value of √(2/9).

Accepted Answer

Introduction / Context: We are asked to estimate a simple radical with a known approximation for √2. Recognizing the algebraic manipulation helps compute quickly without a calculator. Given Data / Assumptions: √2 ≈ 1.4142. We need √(2/9). Concept / Approach: Use √(a/b) = √a / √b. Hence √(2/9) = √2 / 3. Substitute the given approximation √2 ≈ 1.4142 and divide by 3. Step-by-Step Solution: √(2/9) = √2 / 3 ≈ 1.4142 / 31.4142 / 3 ≈ 0.4714 Verification / Alternative check: Square 0.4714: approximately 0.2222, which is close to 2/9 ≈ 0.2222. Why Other Options Are Wrong: 0.2321, 0.3174, 0.4174, and 0.5770 correspond to other unrelated radicals or inverses. Common Pitfalls: Confusing √(2/9) with 1/√(2/9) = 3/√2 (which would be about 2.121), not asked here. Final Answer: 0.4714

Question 8

Given √3 ≈ 1.732, find the approximate value of 1/√3.

Accepted Answer

Introduction / Context: This is a reciprocal radical approximation. With √3 given, compute its reciprocal to three decimal places commonly used in aptitude questions. Given Data / Assumptions: √3 ≈ 1.732. Concept / Approach: 1/√3 = 1 / 1.732. Perform the division to obtain the decimal approximation. Step-by-Step Solution: 1 ÷ 1.732 ≈ 0.577 Verification / Alternative check: Multiply back: 0.577 * 1.732 ≈ 0.999 (close to 1), confirming the approximation. Why Other Options Are Wrong: 0.617 and 0.313 are off the true reciprocal; 0.173 is much too small; 0.707 is 1/√2, not 1/√3. Common Pitfalls: Confusing reciprocals of √2 and √3 due to common values 0.707 and 0.577. Final Answer: 0.577

Question 9

Given √24 ≈ 4.899, compute the exact value of √(8/3) using this information.

Accepted Answer

Introduction / Context: We are to evaluate √(8/3) using the given approximation for √24, observing the algebraic relation between the two expressions. Recognizing equivalent forms simplifies computation. Given Data / Assumptions: √24 ≈ 4.899. We need √(8/3). Concept / Approach: Note that 8/3 = 24/9. Therefore √(8/3) = √(24/9) = √24 / √9 = √24 / 3. Use the provided √24 value and divide by 3. Step-by-Step Solution: √(8/3) = √24 / 3≈ 4.899 / 3 = 1.633 Verification / Alternative check: Square 1.633: approximately 2.666, close to 8/3 ≈ 2.666..., confirming consistency. Why Other Options Are Wrong: 0.544 and 1.333 are too small; 2.666 is the square (not the root); 1.414 is √2, not √(8/3). Common Pitfalls: Confusing √(a/b) with a/b of square roots or squaring mistakenly; always use √(a/b) = √a / √b. Final Answer: 1.633

Question 10

Square-root based division:  Find the value that should replace the question mark so that ? / √2.25 = 550.

Accepted Answer

Introduction / Context: This is a direct square-root and proportion problem. We are given a simple equation involving a square root in the denominator and asked to find the missing numerator so the equality holds. Given Data / Assumptions: Equation: ? / √2.25 = 550. All numbers are real and positive. Arithmetic uses standard real-number operations. Concept / Approach: First, evaluate the square root. Then isolate the unknown by multiplying both sides by that square-root value. This is a single-step linear manipulation once the square root is known. Step-by-Step Solution: Compute √2.25. Since 1.5 * 1.5 = 2.25, √2.25 = 1.5.Given ? / 1.5 = 550.Multiply both sides by 1.5: ? = 550 * 1.5 = 825. Verification / Alternative check: Substitute back: 825 / 1.5 = 550, which confirms the result exactly. Why Other Options Are Wrong: 82.5 and 2 are far too small to yield 550 when divided by 1.5. 3666.66 would give a value much larger than 550 when divided by 1.5. 550 would imply √2.25 = 1, which is incorrect. Common Pitfalls: Forgetting that √2.25 = 1.5, not 1. Also, avoid rounding early—this problem is exact with 1.5. Final Answer: 825

Question 11

Square-root equation with an unknown addend:  If √(75.24 + N) = 8.71, determine the value of N.

Accepted Answer

Introduction / Context: We are reversing a square-root operation to recover the missing addend inside the radical. This is a standard technique in quantitative aptitude involving squaring both sides carefully. Given Data / Assumptions: √(75.24 + N) = 8.71. All quantities are decimal reals. No approximation is required beyond exact decimal squaring. Concept / Approach: Square both sides to eliminate the square root, then isolate N. The arithmetic requires precise decimal squaring of 8.71. Step-by-Step Solution: Square the right-hand side: 8.71^2 = 75.8641.Thus, 75.24 + N = 75.8641.N = 75.8641 − 75.24 = 0.6241. Verification / Alternative check: Check: 75.24 + 0.6241 = 75.8641 and √75.8641 = 8.71 exactly. Why Other Options Are Wrong: 6.241 and 62.41 are off by factors of 10 or 100. “None of these” is incorrect because 0.6241 is listed. 0.06241 is a decimal-place error. Common Pitfalls: Mis-squaring 8.71 or mishandling decimal subtraction can lead to place-value mistakes. Final Answer: .6241

Question 12

Evaluate an exact fractional square root:  Compute √(36.1 / 102.4) and express it as a simplified fraction.

Accepted Answer

Introduction / Context: Converting decimals to exact fractional forms often reveals perfect-square structure. Here, we simplify inside the radical first, then take the square root. Given Data / Assumptions: Expression: √(36.1 / 102.4). Decimals are exact terminating decimals convertible to fractions. Concept / Approach: Multiply numerator and denominator to remove decimals and look for squares. Note that 36.1/102.4 = 361/1024 after scaling by a factor of 10 in numerator and denominator. Step-by-Step Solution: 36.1/102.4 = (361/10) / (1024/10) = 361/1024.Now √(361/1024) = √361 / √1024.√361 = 19, √1024 = 32.Therefore, value = 19/32. Verification / Alternative check: Compute (19/32)^2 = 361/1024 = 36.1/102.4, confirming exactness. Why Other Options Are Wrong: 29/32 and 29/62 have incorrect numerators/denominators. 19/72 does not square to 361/1024. “None of these” is not needed since 19/32 appears. Common Pitfalls: Mis-scaling decimals or taking the square root of 1024 wrongly. Remember 1024 = 2^10, so √1024 = 32. Final Answer: 19/32

Question 13

Radical of a decimal ratio:  Evaluate √(0.289 / 0.00121) and give the exact simplified value.

Accepted Answer

Introduction / Context: Even when decimals look messy, converting to fractional form can expose perfect squares. The goal is to simplify the ratio inside the square root completely before taking the root. Given Data / Assumptions: Compute √(0.289 / 0.00121). Decimals are exact and correspond to familiar squares (289 and 121) after scaling. Concept / Approach: Write decimals as fractions, reduce the ratio, and then take the square root of numerator and denominator separately. Step-by-Step Solution: 0.289 = 289/1000; 0.00121 = 121/100000.Ratio = (289/1000) / (121/100000) = (289 * 100000) / (121 * 1000) = 28900/121.Thus √(0.289 / 0.00121) = √(28900/121) = √28900 / √121.√28900 = √(289 * 100) = 17 * 10 = 170; √121 = 11.Value = 170/11. Verification / Alternative check: Compute (170/11)^2 = 28900/121 = (0.289)/(0.00121), confirming exactness. Why Other Options Are Wrong: 1.7 and 17/10 equal 1.7, far too small. 17/11 ≈ 1.545, also too small. Exact answer is a rational number 170/11. Common Pitfalls: Taking square roots of decimals directly without fraction conversion; misplacing decimal points in scaling. Final Answer: 170/11

Question 14

Equalizing radicals by squaring:  Solve for ? if √(25/15625) = √(?/30625).

Accepted Answer

Introduction / Context: When two square roots are equal, their radicands (nonnegative) must be equal. We simplify one side and equate the interior fractions to find the missing numerator. Given Data / Assumptions: √(25/15625) = √(?/30625). All quantities are nonnegative. Concept / Approach: First simplify 25/15625. Then set the simplified fraction equal to ?/30625 and solve for the unknown numerator. Step-by-Step Solution: 25/15625 = 1/625 (divide numerator and denominator by 25).So √(25/15625) = √(1/625) = 1/25.Given equality of square roots ⇒ ?/30625 = 1/625.Cross-multiply: ? = 30625 / 625 = 49. Verification / Alternative check: Compute √(?/30625) with ? = 49: 49/30625 = 1/625; √(1/625) = 1/25, matching the left side. Why Other Options Are Wrong: 2, 35, 1225, 625 do not produce the same radicand 1/625 on the right side. Common Pitfalls: Equating the square roots without equating the radicands, or forgetting to simplify 25/15625 first. Final Answer: 49

Question 15

Solve for the radicand from a quotient:  Find ? such that 1872 / √? = 234.

Accepted Answer

Introduction / Context: This problem asks for the number whose square root appears in a denominator. It reduces to isolating the square root and then squaring to retrieve the radicand. Given Data / Assumptions: Equation: 1872 / √? = 234. All values are positive reals. Concept / Approach: Isolate √? by dividing 1872 by 234. Once √? is known, square it to find ?. Step-by-Step Solution: Compute √? = 1872 / 234.234 * 8 = 1872, so √? = 8.Square both sides: ? = 8^2 = 64. Verification / Alternative check: Check: 1872 / √64 = 1872 / 8 = 234. Works exactly. Why Other Options Are Wrong: 324, 256 would yield square roots 18 or 16, not 8. 8 is √?, not ? itself. 36 gives √? = 6, which is incorrect here. Common Pitfalls: Stopping at √? = 8 and selecting 8 as the answer. The question asks for ?, not its square root. Final Answer: 64

Question 16

Clarified grouping of radicals and divisors:  Compute (√324) / 1.5 = ? / √256 and find the value of ?.

Accepted Answer

Introduction / Context: The original expression can be ambiguous without parentheses. Interpreting it as (√324) / 1.5 = ? / √256 matches the provided numeric options. We solve for the unknown numerator. Given Data / Assumptions: (√324)/1.5 = ? / √256. √324 = 18; √256 = 16. Concept / Approach: Evaluate both square roots, compute the left-hand side, then cross-multiply to find ?. Step-by-Step Solution: Left-hand side: (√324)/1.5 = 18 / 1.5 = 12.Right-hand side: ? / 16.Set equal: 12 = ?/16 ⇒ ? = 12 * 16 = 192. Verification / Alternative check: Substitute back: 192 / 16 = 12, which equals 18 / 1.5. Consistent. Why Other Options Are Wrong: 432, 288, 122, 64 do not satisfy ? / 16 = 12. Common Pitfalls: Misreading the original as √(324/1.5), which would introduce an irrational factor not present in the options. Final Answer: 192

Question 17

Nested decimal radicals simplification:  Evaluate √[(0.324 × 0.081 × 4.624) / (1.5625 × 0.0289 × 72.9 × 64)].

Accepted Answer

Introduction / Context: This expression looks complicated but becomes simple when decimals are written as exact fractions and factors are canceled before taking the square root. Given Data / Assumptions: Numerator: 0.324, 0.081, 4.624. Denominator: 1.5625, 0.0289, 72.9, 64. All are exact decimals (e.g., 1.5625 = 25/16; 0.0289 = 17^2/10^4). Concept / Approach: Convert to prime-powered fractions, cancel common factors systematically, then take the square root at the end to avoid rounding. Step-by-Step Solution: 0.324 = 81/250; 0.081 = 81/1000; 4.624 = 578/125 = 2 * 17^2 / 5^3.1.5625 = 25/16 = 5^2/2^4; 0.0289 = 17^2/10^4; 72.9 = 729/10 = 3^6/(2*5); 64 = 2^6.After cancellations, the inside reduces exactly to 9/15625.Therefore, the value is √(9/15625) = 3/125 = 0.024. Verification / Alternative check: Compute numerically with high precision: result ≈ 0.02400, confirming the exact fractional outcome. Why Other Options Are Wrong: 24 and 2.40 are off by factors of 1000 and 100. “None of these” is unnecessary since 0.024 matches exactly. 0.12 is five times too large. Common Pitfalls: Rounding mid-steps instead of canceling exactly; mixing percent and decimal interpretations. Final Answer: 0.024

Question 18

Sum of square roots with scaled decimals:  Given √15625 = 125, evaluate √15625 + √156.25 + √1.5625.

Accepted Answer

Introduction / Context: We combine square roots of numbers that are decimal scalings of one another. Recognizing powers of 10 under square roots simplifies computation. Given Data / Assumptions: √15625 = 125 (given). 156.25 = 15625 / 100. 1.5625 = 15625 / 10000. Concept / Approach: Use √(A/100) = √A / 10 and √(A/10000) = √A / 100 for positive A. Apply to A = 15625. Step-by-Step Solution: √15625 = 125.√156.25 = √(15625/100) = √15625 / 10 = 125 / 10 = 12.5.√1.5625 = √(15625/10000) = √15625 / 100 = 125 / 100 = 1.25.Sum = 125 + 12.5 + 1.25 = 138.75. Verification / Alternative check: Compute squares: 12.5^2 = 156.25 and 1.25^2 = 1.5625, confirming the decomposition. Why Other Options Are Wrong: Values like 1.3875 and 13.875 drop decimal places incorrectly. 156.25 confuses a square root with the original radicand. Common Pitfalls: Forgetting to divide by 10 or 100 when taking square roots of numbers reduced by factors of 100 or 10000. Final Answer: 138.75

Question 19

Proportion with two radicals:  Find the positive value of ? such that √1296 / ? = ? / 2.25.

Accepted Answer

Introduction / Context: This is a symmetric proportion where the unknown appears in numerator and denominator. Recognize the product-of-means equals product-of-extremes structure to form a perfect square in the unknown. Given Data / Assumptions: Equation: √1296 / x = x / 2.25. x is positive (since it represents a length-like quantity under a square root balance). Concept / Approach: Cross-multiply to obtain x^2 = (√1296) * 2.25. Then evaluate the square root and multiply. Step-by-Step Solution: √1296 = 36.Cross-multiply: x^2 = 36 * 2.25.2.25 = 9/4 ⇒ 36 * 2.25 = 81.x = √81 = 9 (positive root). Verification / Alternative check: Check: Left side = 36/9 = 4; right side = 9/2.25 = 4. They match. Why Other Options Are Wrong: 6, 7, 8 give unequal sides when substituted. 4.5 is half the correct value and fails the equality. Common Pitfalls: Taking √1296 incorrectly or forgetting only the positive root makes sense in this proportion context. Final Answer: 9

Question 20

Square root of a small decimal:  Compute √0.00059049 and select the exact decimal value.

Accepted Answer

Introduction / Context: Recognizing digit patterns helps here: 59049 is a familiar square (243^2). We only need to account for decimal places correctly when taking the square root. Given Data / Assumptions: Number: 0.00059049. We know 243^2 = 59049. Concept / Approach: If a^2 = 59049, then (a/10^k)^2 = 0.00059049 for the appropriate k. Count decimal places to determine the square-root shift. Step-by-Step Solution: 59049 has 5 digits; here 0.00059049 = 59049 / 10^8.√(59049 / 10^8) = 243 / 10^4 = 0.0243.Therefore, √0.00059049 = 0.0243. Verification / Alternative check: (0.0243)^2 = 0.00059049 (since 243^2 = 59049 and (10^4)^2 = 10^8). Why Other Options Are Wrong: .243 squares to 0.059049 (ten times too large). .00243 squares to 0.0000059049 (ten times too small). .000243 is off by two decimal places for the square root. Common Pitfalls: Miscalculating decimal-place shifts when taking square roots. Count digits carefully. Final Answer: .0243

Square Root and Cube Root Questions