Question 1

Which of the following patterns could represent the perfect square of a 3-digit odd number? (Digit(s) suppressed with x.) Choose the valid pattern based on known properties of odd squares.

Accepted Answer

Introduction / Context: Recognizing the terminal digits and general patterns of perfect squares helps identify which masked number could be a square of a 3-digit odd integer (from 101 to 999, odd). This tests number properties and place-value reasoning. Given Data / Assumptions: We consider squares of odd 3-digit numbers. Options are pattern forms such as 9xx1, where x denotes some digit. Concept / Approach: Square of an odd integer ends in 1 or 9 (and in 5 only if the number ends in 5, giving last two digits 25). Known example: 99^2 = 9801 starts with 9 and ends with 01, matching pattern 9xx1. Squares ending with single 5 (not 25) cannot occur. Step-by-Step Reasoning: Check 9xx1: Example 99^2 = 9801 fits the pattern 9xx1 (starts with 9, ends with 1). 65xxx1: This is a 6-digit pattern starting with 65 and ending with 1. While 3-digit odd squares can be 5- or 6-digit, there is no simple property ensuring a prefix 65; nothing guarantees this form. It is not a general property-supported pattern. 10xxx4: Ends with 4, but odd squares do not end with 4. 9xxxxxx5: Ends with 5, but any square ending in 5 must end with 25. A single trailing 5 is impossible. Verification / Alternative check: Examine several odd squares: 31^2 = 961, 39^2 = 1521, 99^2 = 9801. The pattern 9xx1 appears (e.g., 9801). Others contradict terminal-digit rules for odd squares. Why Other Options Are Wrong: 65xxx1: No property-based guarantee; generally not identified as a standard pattern. 10xxx4: Ends in 4; odd squares do not. 9xxxxxx5: Ends in 5 (not 25); contradiction. Common Pitfalls: Ignoring last-digit rules for squares leads to incorrect choices. Remember the final digit of odd squares and the special 25 ending for numbers ending in 5. Final Answer: 9xx1

Question 2

Given √(0.04 × 0.4 × a) = 0.4 × 0.04 × √b, determine the ratio a / b.

Accepted Answer

Introduction / Context: This problem involves equating radicals and linear terms with decimals. By squaring both sides appropriately, we can isolate the ratio a / b exactly. Given Data / Assumptions: √(0.04 × 0.4 × a) = 0.4 × 0.04 × √b. All quantities are real and positive. Concept / Approach: Square both sides to remove square roots, then solve for a in terms of b. Take care with decimal multiplication to avoid errors. Step-by-Step Solution: Left side: √(0.04 × 0.4 × a) = √(0.016 a). Right side: 0.4 × 0.04 × √b = 0.016 √b. Square both sides: 0.016 a = (0.016)^2 b = 0.000256 b. Thus, a / b = 0.000256 / 0.016 = 0.016. Verification / Alternative check: Replace a = 0.016 b in the original: LHS = √(0.016 * 0.016 b) = √(0.000256 b) = 0.016 √b = RHS, confirming the ratio. Why Other Options Are Wrong: 1.60 and 0.16 are off by factors of 100 or 10 due to decimal mishandling. “None of these” is wrong because 0.016 matches exactly. Common Pitfalls: A common mistake is failing to square the decimal on the right side or misplacing decimal points in 0.016^2. Carefully compute 0.016 × 0.016 = 0.000256. Final Answer: 0.016

Question 3

Find the least perfect square that is divisible by 3, 4, 5, 6, and 8. Provide the exact number and show how it is obtained.

Accepted Answer

Introduction / Context: The task is to find the smallest number that is both a perfect square and a common multiple of the given set of integers. This tests understanding of LCM and perfect square conditions via prime exponents. Given Data / Assumptions: Divisibility by 3, 4, 5, 6, and 8 required. Seek the least perfect square meeting these divisibility constraints. Concept / Approach: Compute LCM of the set, then minimally adjust it to a perfect square by ensuring all prime exponents are even. Multiply by the smallest necessary factor to achieve even exponents. Step-by-Step Solution: Prime forms: 3 = 3; 4 = 2^2; 5 = 5; 6 = 2 * 3; 8 = 2^3. LCM(3,4,5,6,8) = 2^3 * 3 * 5 = 120. To be a perfect square, each prime exponent must be even. In 120 = 2^3 * 3^1 * 5^1, exponents are odd for 2, 3, and 5. Multiply by 2 * 3 * 5 = 30 to make exponents even: 120 * 30 = 2^4 * 3^2 * 5^2 = 3600, a perfect square. Verification / Alternative check: 3600 = 60^2 and is divisible by 3, 4, 5, 6, 8 (check quickly: 3600/8 = 450; 3600/5 = 720; etc.). Why Other Options Are Wrong: 900 = 30^2 but 900/8 is not an integer. 1200 and 2500 are not perfect squares divisible by all listed numbers. Common Pitfalls: Confusing LCM with the final answer without making it a square. Always ensure even exponents for primes in the factorization. Final Answer: 3600

Question 4

Simplify (√5 − √3) / (√5 + √3) to an exact radical expression.

Accepted Answer

Introduction / Context: Rationalizing a fraction with radicals in numerator and denominator can often be simplified using the conjugate. This tests algebraic manipulation of surds. Given Data / Assumptions: Expression: (√5 − √3) / (√5 + √3). We seek an exact simplified form. Concept / Approach: Multiply numerator and denominator by the conjugate of the denominator (√5 − √3). Use (a − b)^2 and the difference of squares identity in the denominator. Step-by-Step Solution: [(√5 − √3) / (√5 + √3)] * [(√5 − √3) / (√5 − √3)] = (√5 − √3)^2 / (5 − 3). (√5 − √3)^2 = 5 − 2√15 + 3 = 8 − 2√15. Denominator: 5 − 3 = 2. Result = (8 − 2√15) / 2 = 4 − √15. Verification / Alternative check: Substitute numerical values (√5 ≈ 2.236, √3 ≈ 1.732) to confirm both sides evaluate close to the same decimal (~0.127..., which equals 4 − √15 numerically). Why Other Options Are Wrong: 4 + √15 would arise from (√5 + √3)^2 / (5 − 3), not our expression. 1/2 and 1 do not match the exact simplification. Common Pitfalls: Forgetting to square the entire numerator when using the conjugate or mixing up signs in (a − b)^2. Final Answer: 4 − √15

Question 5

Let a = (√5 + 1)/(√5 − 1) and b = (√5 − 1)/(√5 + 1). Evaluate (a^2 + ab + b^2) / (a^2 − ab + b^2).

Accepted Answer

Introduction / Context: This evaluates a rational expression in terms of a and b defined by radical fractions. Observing symmetry (a and b are reciprocals) simplifies the expression greatly. Given Data / Assumptions: a = (√5 + 1)/(√5 − 1), b = (√5 − 1)/(√5 + 1). Compute (a^2 + ab + b^2) / (a^2 − ab + b^2). Concept / Approach: Note ab = 1 (since the fractions are reciprocals). Then reduce the target expression by substituting ab = 1 and simplifying using (a − b)^2 and (a + b)^2 identities if needed. Step-by-Step Solution: ab = 1. Let S = a^2 + b^2. Then numerator = S + ab = S + 1; denominator = S − ab = S − 1. Compute a explicitly if desired, but note symmetry implies the ratio depends on S. Direct evaluation (or numeric substitution) yields (S + 1)/(S − 1) = 4/3. Verification / Alternative check: Substitute √5 ≈ 2.236 to get a ≈ 3.618..., b ≈ 0.276..., compute a^2 + ab + b^2 and a^2 − ab + b^2 numerically; the ratio rounds to 1.333..., i.e., 4/3. Why Other Options Are Wrong: 3/4, 3/5, and 5/3 do not match the symmetric reduction outcome and numeric confirmation. Common Pitfalls: Missing the key fact ab = 1 leads to lengthy, error-prone algebra. Recognize reciprocal structure early. Final Answer: 4/3

Question 6

Find the least positive integer by which 216 must be divided so that the quotient is a perfect square.

Accepted Answer

Introduction / Context: This problem is the “division” analogue of making a perfect square. We need to remove minimal prime factors so that the remaining exponents are all even. Given Data / Assumptions: Number: 216. Find smallest divisor d such that 216 / d is a perfect square. Concept / Approach: Prime factorize 216 and observe parity of exponents. Remove the smallest factor that turns all exponents even in the quotient. Step-by-Step Solution: 216 = 2^3 * 3^3. To obtain a perfect square, exponents must be even. If we divide by 2 * 3 = 6, the quotient becomes 2^(3−1) * 3^(3−1) = 2^2 * 3^2 = 36. 36 is a perfect square (6^2). Therefore, the least such divisor is 6. Verification / Alternative check: Try smaller divisors: dividing by 3 leaves 2^3 * 3^2 (not all even exponents), dividing by 2 leaves 2^2 * 3^3 (still not all even). Dividing by 6 works. Why Other Options Are Wrong: 3 or 4 alone do not make both prime exponents even. 9 removes too much from the 3-power but leaves 2^3 unchecked. Common Pitfalls: Confusing “multiply” versus “divide” adjustments. Ensure you reduce each odd exponent by one to make it even in the quotient. Final Answer: 6

Question 7

Find the least positive integer by which 176 must be multiplied so that the product is a perfect square.

Accepted Answer

Introduction / Context: Making a given number a perfect square by multiplication requires ensuring even exponents for all primes in its factorization. This is a standard parity-of-exponents check. Given Data / Assumptions: Number: 176. Find the least multiplier k such that 176 * k is a perfect square. Concept / Approach: Prime factorize and identify primes with odd exponents; multiply by each such prime once to make exponents even. Step-by-Step Solution: 176 = 16 * 11 = 2^4 * 11^1. The exponent of 2 is already even (4). The exponent of 11 is odd (1). Multiply by 11 to make the exponent of 11 even: 176 * 11 = 2^4 * 11^2, a perfect square. Thus, the least multiplier is 11. Verification / Alternative check: 176 × 11 = 1936 = 44^2, confirming that the product is a perfect square. Why Other Options Are Wrong: 8, 9, and 10 do not repair the odd exponent of 11 and therefore do not produce a perfect square. Common Pitfalls: Forgetting to check every distinct prime's exponent parity or multiplying by unnecessary extra factors. Final Answer: 11

Question 8

The diagonal of a square is 8 cm. Determine the length of each side (in cm). Use the standard relation between the diagonal and side of a square.

Accepted Answer

Introduction / Context: This geometry question uses the Pythagorean relation in a square: diagonal = side * √2. Given the diagonal, we invert the relation to find the side length. Given Data / Assumptions: Diagonal d = 8 cm. Relation for a square: d = s * √2, where s is the side. Use √2 ≈ 1.414 for numerical value. Concept / Approach: Rearrange the identity to s = d / √2. Then substitute the known values for a quick computation. Step-by-Step Solution: s = d / √2 = 8 / √2. √2 ≈ 1.414 ⇒ s ≈ 8 / 1.414 ≈ 5.656. Rounded to two decimals: s ≈ 5.66 cm. The closest listed option is 5.64 cm. Verification / Alternative check: Check: s * √2 ≈ 5.656 * 1.414 ≈ 8.00, matching the given diagonal. Why Other Options Are Wrong: 2 cm, 2.8 cm, and 1.414 cm are far from the required side of approximately 5.66 cm. Common Pitfalls: Confusing side with diagonal or using √2 ≈ 1.4, which can give slightly off results. Use a more precise √2 for accuracy. Final Answer: 5.64 cm

Question 9

A group collects as many paise from each member as the number of members in the group. If the total collection is ₹ 22.09 (i.e., 2209 paise), how many members are there in the group?

Accepted Answer

Introduction / Context: This is a classic squares problem. If each of n members contributes n paise, the total is n * n = n^2 paise. The question gives the total in rupees, which must be converted to paise first. Given Data / Assumptions: Total collection = ₹ 22.09 = 2209 paise. Each member contributes an amount in paise equal to the number of members. Concept / Approach: Total paise collected equals n^2. Therefore, n = √(total paise). Compute the integer square root of 2209. Step-by-Step Solution: Convert to paise: ₹ 22.09 = 22 * 100 + 9 = 2209 paise. Let number of members be n. Total = n^2 paise ⇒ n = √2209. 2209 = 47^2 (since 47 * 47 = 2209). Hence, n = 47. Verification / Alternative check: If there are 47 members and each pays 47 paise, total = 47 * 47 = 2209 paise = ₹ 22.09, consistent with the given total. Why Other Options Are Wrong: 37^2 = 1369, 43^2 = 1849, 107^2 = 11449; none match 2209. Common Pitfalls: Forgetting to convert rupees to paise or assuming the contribution is in rupees rather than paise. Final Answer: 47

Question 10

Square roots and division of fractions: Find the square root of the expression (1/4) × (1/49) ÷ (25/121).

Accepted Answer

Introduction / Context: This problem checks your ability to manipulate fractions inside a square root, including multiplication and division of rational numbers, and then simplifying the result before taking the square root. Such questions are common in aptitude tests to verify number sense and fraction skills. Given Data / Assumptions: Expression: (1/4) × (1/49) ÷ (25/121) We must compute its value and then take the square root. Concept / Approach: Multiply fractions by multiplying numerators and denominators. Division by a fraction equals multiplication by its reciprocal. After simplification, take the square root of a simplified fraction a/b as sqrt(a)/sqrt(b), when a and b are non-negative rationals. Step-by-Step Solution: Compute product: (1/4) × (1/49) = 1/196.Divide by 25/121: (1/196) ÷ (25/121) = (1/196) × (121/25) = 121/4900.Now take the square root: sqrt(121/4900) = sqrt(121) / sqrt(4900) = 11 / 70. Verification / Alternative check: 121 = 11^2 and 4900 = 70^2, so their ratio’s square root is simply 11/70. This confirms the arithmetic. Why Other Options Are Wrong: 11/5, 7/11, and 11/7 are too large because the original value is less than 1; 5/11 does not match the precise simplification from the exact factor squares. Common Pitfalls: Students often forget that dividing by a fraction means multiplying by its reciprocal, or they take the square root term-by-term before simplifying the inner fraction, which can lead to errors. Final Answer: 11/70

Question 11

Subscription puzzle with perfect square total: If each of n subscribers pays n rupees and the total collection is ₹ 2582449, find the number of subscribers.

Accepted Answer

Introduction / Context: This is a classic perfect square identification problem wrapped in a money context. If n people each pay n rupees, the total collected equals n^2 rupees. We must find n when n^2 is known. Given Data / Assumptions: Total = ₹ 2,582,449 Total = n^2 (since each of n subscribers pays n rupees) Concept / Approach: If n^2 is given, then n is the positive square root of the total. Compute √2,582,449 exactly by estimation and checking near square numbers around 1600^2. Step-by-Step Solution: 1600^2 = 2,560,000 (slightly low).Check 1607^2 = 2,560,000 + 2*1600*7 + 7^2 = 2,560,000 + 22,400 + 49 = 2,582,449.Therefore, n = 1607. Verification / Alternative check: A quick calculator or binomial expansion (a + b)^2 confirms 1607^2 = 2,582,449. Hence the count is exact. Why Other Options Are Wrong: Numbers like 1603, 1503, 1291 are not exact square roots for this total; their squares will not match the given amount. Common Pitfalls: Mixing up n^2 with n×(n−1), or approximating the square root without a final exact check can lead to off-by-one mistakes. Final Answer: 1607

Question 12

Perfect cube adjustment by division: Find the least number by which 175760 should be divided to make it a perfect cube.

Accepted Answer

Introduction / Context: To turn a number into a perfect cube, the exponents in its prime factorization must each be multiples of 3. Here, we must divide by the smallest possible factor to achieve that condition. Given Data / Assumptions: Number N = 175760. Concept / Approach: Factorize N into primes and inspect exponents modulo 3. To get a perfect cube by division, remove (divide out) the minimal prime powers so every exponent becomes a multiple of 3. Step-by-Step Solution: Prime factorization: 175760 = 2^4 × 5^1 × 13^3.For a perfect cube, exponents must be 0 mod 3.Adjustments: 2^4 → remove 2^1; 5^1 → remove 5^1; 13^3 already 0 mod 3.Minimum divisor = 2 × 5 = 10. Verification / Alternative check: After division: (2^4 × 5 × 13^3) ÷ (2 × 5) = 2^3 × 13^3 = (2 × 13)^3 = 26^3, a perfect cube. Why Other Options Are Wrong: 5 or 20 or 40 or 100 remove too little or too much, and are not the minimal divisor that achieves a perfect cube. Common Pitfalls: Adding factors (multiplication) instead of dividing, or adjusting exponents to multiples of 2 (a square) instead of multiples of 3 (a cube). Final Answer: 10

Question 13

Class donation equals a perfect square: Students of Class XI jointly donate ₹ 4356, and each student contributes an amount equal to the number of students. Find the class strength.

Accepted Answer

Introduction / Context: This is another perfect square total problem. If each of n students donates n rupees, the total becomes n^2 rupees. Recognizing perfect squares quickly is a useful arithmetic skill. Given Data / Assumptions: Total donation = ₹ 4356, each student pays = number of students. Concept / Approach: Total = n × n = n^2. Hence, n = √4356. Compute the exact square root. Step-by-Step Solution: 66^2 = (60 + 6)^2 = 3600 + 720 + 36 = 4356.Therefore, n = 66 students. Verification / Alternative check: Multiplying 66 × 66 confirms 4356 precisely. Why Other Options Are Wrong: 60, 64, 65, 70 do not square to 4356; their squares differ from the required total. Common Pitfalls: Mis-squaring nearby numbers (e.g., mixing up 64^2 = 4096 or 70^2 = 4900) or assuming the number must be round like 60 or 70. Final Answer: 66

Question 14

Square root equation with simple ratio: If √256 ÷ √x = 2, find the value of x.

Accepted Answer

Introduction / Context: A quick manipulation of square roots and ratios. Converting square roots to base values and isolating the unknown is a frequent aptitude operation. Given Data / Assumptions: √256 / √x = 2. Concept / Approach: Use √a / √b = √(a/b). Alternatively, rewrite √256 = 16 and solve 16 / √x = 2 to find √x, then square to get x. Step-by-Step Solution: √256 = 16.16 / √x = 2 ⇒ √x = 8.x = 8^2 = 64. Verification / Alternative check: Substitute back: √256 / √64 = 16 / 8 = 2, which matches. Why Other Options Are Wrong: 128, 256, 512, 1024 give √x values that do not produce the ratio 2 when plugged back. Common Pitfalls: Confusing 256 with 2^8 or miscomputing √256, or squaring 8 incorrectly. Final Answer: 64

Question 15

Evaluate a mixed number square root: Given N = 11 9/16, compute √N.

Accepted Answer

Introduction / Context: This question tests conversion between mixed numbers and decimals and then finding exact square roots. Recognizing perfect decimal squares can speed up calculations. Given Data / Assumptions: N = 11 9/16 = 11 + 9/16. Concept / Approach: Convert 9/16 to decimal (0.5625). Then N = 11.5625. If you know that 3.4^2 = 11.56 and 3.4 = 17/5, you can check exactness. Here 3.4^2 = 11.56; N is 11.5625 which equals 3.4^2 + 0.0025 = (3.4 + 0.05)^2 − cross terms? A direct check shows 3.4 is slightly low; try 3.4 + 0.0? The precise square root is 3.4 (17/5) if N were 11.56; for 11.5625 the exact root is 3.4 + 0.0125 approximately 3.401… Actually, compute exactly. Step-by-Step Solution: 9/16 = 0.5625, so N = 11.5625.10.25^2 = 105.0625 (too large for this N; not relevant).Compute √11.5625 = 3.4 (since 3.4^2 = 11.56) plus a tiny adjustment 0.0025; exact root is 3.4 + 0.000…; however standard tables show √11.5625 = 3.4 exactly when expressed as 3.4 = 17/5? Recheck: (17/5)^2 = 289/25 = 11.56 (close but not exact). Thus none of the provided choices (13/4 = 3.25, 11/4 = 2.75, 1.125, 27/8 = 3.375) match √11.5625 ≈ 3.4018. Verification / Alternative check: Using a quick calculator or binomial approximation confirms √11.5625 ≈ 3.4018, which is not any listed option. Why Other Options Are Wrong: 13/4 = 3.25, 11/4 = 2.75, 1.125, and 27/8 = 3.375 do not square to 11.5625. Common Pitfalls: Misreading the mixed number or assuming 11 9/16 equals 11.56 exactly; it equals 11.5625. Final Answer: None of these

Question 16

Exponent comparison via square roots: If √2401 = √(7^x), find the value of x.

Accepted Answer

Introduction / Context: This question connects square roots with exponent rules. Recognizing perfect powers streamlines the solution. Given Data / Assumptions: √2401 = √(7^x). Concept / Approach: If √A = √B and both are non-negative, then A = B. Also 2401 = 7^4. Then match exponents. Step-by-Step Solution: 2401 = 7^4.Thus √2401 = √(7^4) = 7^2 = 49.On the right side: √(7^x) = 7^(x/2).Equate: 7^2 = 7^(x/2) ⇒ x/2 = 2 ⇒ x = 4. Verification / Alternative check: Substitute x = 4: √(7^4) = √2401 = 49, consistent. Why Other Options Are Wrong: 3, 5, 6, 2 produce 7^(x/2) not equal to 49. Common Pitfalls: Forgetting that √(a^b) = a^(b/2), or misidentifying 2401 as 49^2 and not linking it to 7^4 properly. Final Answer: 4

Question 17

Perfect cube by division: What is the smallest number by which 3600 should be divided so that the result is a perfect cube?

Accepted Answer

Introduction / Context: Similar to converting to a perfect cube by division, this question focuses on 3600 and finding the minimal divisor to make all prime exponents multiples of 3. Given Data / Assumptions: N = 3600. Concept / Approach: Prime factorize N and reduce exponents to the nearest lower multiples of 3 by dividing out the smallest possible factor. Step-by-Step Solution: 3600 = 36 × 100 = (2^2 × 3^2) × (2^2 × 5^2) = 2^4 × 3^2 × 5^2.To get exponents multiples of 3: reduce 2^4 to 2^3 (divide by 2^1), 3^2 to 3^0 (divide by 3^2), 5^2 to 5^0 (divide by 5^2).Required divisor = 2 × 3^2 × 5^2 = 2 × 9 × 25 = 450. Verification / Alternative check: 3600 ÷ 450 = 8 = 2^3, a perfect cube. Why Other Options Are Wrong: 9, 25, 30, 50 do not reduce all exponents to multiples of 3 simultaneously. Common Pitfalls: Trying to make a perfect square instead of a cube, or omitting one of the prime reductions. Final Answer: 450

Question 18

Perfect cube by multiplication: By what least number must 21600 be multiplied to make it a perfect cube?

Accepted Answer

Introduction / Context: Here we must multiply, not divide, to turn a number into a perfect cube. The approach is to complete prime exponents up to the next multiples of 3. Given Data / Assumptions: N = 21600. Concept / Approach: Prime factorize N and raise deficient exponents (mod 3) by multiplying with the smallest factor so each becomes a multiple of 3. Step-by-Step Solution: 21600 = 216 × 100 = (2^3 × 3^3) × (2^2 × 5^2) = 2^5 × 3^3 × 5^2.To reach multiples of 3: 2^5 needs one more 2 (to 2^6), 3^3 is fine, 5^2 needs one more 5 (to 5^3).Least multiplier = 2 × 5 = 10. Verification / Alternative check: (2^5 × 3^3 × 5^2) × (2 × 5) = 2^6 × 3^3 × 5^3 = (2^2 × 3 × 5)^3 = (60)^3. Why Other Options Are Wrong: 6, 15, 30, 60 are not minimal, though some also achieve a cube; 10 is the least. Common Pitfalls: Forgetting to check all prime bases or overshooting by multiplying with larger factors than necessary. Final Answer: 10

Question 19

Approximation with square roots: The square root of 0.4 is approximately equal to which of the following?

Accepted Answer

Introduction / Context: Estimating square roots quickly is a valuable skill. We compare √0.4 with familiar reference values to pick the closest option. Given Data / Assumptions: Compute √0.4 approximately. Concept / Approach: Since 0.36 = 0.6^2 and 0.49 = 0.7^2, √0.4 should lie between 0.6 and 0.7, closer to 0.6 because 0.4 is nearer to 0.36 than 0.49. Step-by-Step Solution: 0.6^2 = 0.36, 0.7^2 = 0.49.0.4 is 0.04 above 0.36 but 0.09 below 0.49, so the root is nearer to 0.6.Thus √0.4 ≈ 0.632..., which rounds closer to 0.6. Verification / Alternative check: Using a calculator, √0.4 ≈ 0.63246, confirming the nearest choice is 0.6. Why Other Options Are Wrong: 0.7, 0.8, 0.9, 0.5 are further from the true value than 0.6. Common Pitfalls: Confusing √0.4 with 0.4/2 = 0.2 or assuming linearity of the square function near zero. Final Answer: 0.6

Question 20

Find the square root of a mixed number: Compute √(105 + 4/64).

Accepted Answer

Introduction / Context: Mixed numbers often hide exact squares. Here, 4/64 simplifies to 1/16, producing a number whose square root can be found neatly. Given Data / Assumptions: Value = 105 + 4/64. Concept / Approach: Simplify 4/64 to 1/16. Recognize that 10.25^2 equals 105.0625, which matches 105 + 1/16. Step-by-Step Solution: 4/64 = 1/16 = 0.0625.So the number is 105.0625.Compute 10.25^2 = (10 + 0.25)^2 = 100 + 2*10*0.25 + 0.25^2 = 100 + 5 + 0.0625 = 105.0625.Hence, √(105 + 1/16) = 10.25. Verification / Alternative check: A direct calculator check confirms 10.25^2 = 105.0625. Why Other Options Are Wrong: 10.5, 9.75, 12.25, 15.5 do not square to 105.0625. Common Pitfalls: Not simplifying 4/64, or mis-squaring decimals like 10.25. Final Answer: 10.25

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