Question 1

Evaluate the fractional power precisely: Compute (0.00032)^(2/5) and express your answer as a simple fraction.

Accepted Answer

Introduction / Context:This question checks fluency with fractional exponents and scientific notation. Rewriting decimals as powers of 10 and small integers as powers of 2 helps simplify the evaluation of (0.00032)^(2/5). Given Data / Assumptions: Decimal: 0.00032 Exponent: 2/5 Answer required as a reduced fraction. Concept / Approach:Express 0.00032 as a ratio using prime bases. Note 0.00032 = 32/100000 with 32 = 2^5 and 100000 = 10^5. Then apply (a/b)^(2/5) = a^(2/5) / b^(2/5) and use exponent rules to collapse powers cleanly. Step-by-Step Solution:0.00032 = 32 / 100000 = 2^5 / 10^5(0.00032)^(2/5) = (2^5 / 10^5)^(2/5)= 2^(5*(2/5)) / 10^(5*(2/5))= 2^2 / 10^2 = 4 / 100 = 1/25 Verification / Alternative check:Compute the fifth root first: (0.00032)^(1/5) = 0.2 (since 0.2^5 = 0.00032), then square: 0.2^2 = 0.04 = 1/25. Why Other Options Are Wrong:1/125 and 1/225 arise from misapplying the root or squaring steps; 1/625 would be (1/5)^4, not applicable here; 1/20 is a decimal rounding error. Common Pitfalls:Treating 0.00032 as 3.2×10^-4 instead of 3.2×10^-4 is fine, but ensure consistent exponent distribution; the prime-power route avoids slips. Final Answer:1/25

Question 2

Solve the exponential sum: If 2^(x−1) + 2^(x+1) = 2560, find the value of x.

Accepted Answer

Introduction / Context:This evaluates comfort with factoring common powers in exponential equations. Recognizing a shared factor in 2^(x−1) and 2^(x+1) simplifies the equation greatly. Given Data / Assumptions: 2^(x−1) + 2^(x+1) = 2560 2 is the common base. Concept / Approach:Factor out the smaller power 2^(x−1). Use 2^(x+1) = 2^(x−1)*2^2 to combine terms. Then solve a simple power-of-two equation by matching exponents. Step-by-Step Solution:2^(x−1) + 2^(x+1) = 2^(x−1)(1 + 2^2) = 2^(x−1)*5So 2^(x−1)*5 = 2560 ⇒ 2^(x−1) = 2560 / 5 = 512512 = 2^9 ⇒ x − 1 = 9 ⇒ x = 10 Verification / Alternative check:Check: 2^9 + 2^11 = 512 + 2048 = 2560, correct. Why Other Options Are Wrong:Values like 8, 9, 11, 12 do not satisfy the exact power equality when substituted. Common Pitfalls:Factoring out 2^x instead of 2^(x−1) is fine but gives fractions; choosing the smallest exponent keeps arithmetic clean. Final Answer:10

Question 3

Apply index laws: Evaluate (10)^(200) ÷ (10)^(196) and report the exact numerical value.

Accepted Answer

Introduction / Context:This is a direct application of exponent subtraction for division with the same base. The goal is to demonstrate fluency with a^m / a^n = a^(m−n). Given Data / Assumptions: (10)^(200) ÷ (10)^(196) Base is identical (10). Concept / Approach:Use the law a^m / a^n = a^(m−n). Subtract exponents and then evaluate the resulting power of 10. Step-by-Step Solution:(10)^(200) ÷ (10)^(196) = 10^(200 − 196) = 10^410^4 = 10000 Verification / Alternative check:Think place value: shifting decimal four places to the right matches multiplying by 10000. Why Other Options Are Wrong:1000 (10^3), 100 (10^2), or 100000 (10^5) correspond to incorrect exponent differences. 10 is 10^1. Common Pitfalls:Accidentally adding exponents during division. Remember division subtracts exponents. Final Answer:10000

Question 4

Nested surds identity (repaired for clarity): Evaluate √(5 + 2√6) − 1 / √(5 − 2√6).

Accepted Answer

Introduction / Context:This classic surds problem relies on recognizing conjugate forms generated by binomial squares. The original database text was ambiguous; applying the Recovery-First Policy, we clarify it as the well-known identity: √(5 + 2√6) − 1 / √(5 − 2√6). Given Data / Assumptions: Expression: √(5 + 2√6) − 1/√(5 − 2√6) All radicals are principal square roots. Concept / Approach:Notice (√3 + √2)^2 = 3 + 2 + 2√6 = 5 + 2√6, so √(5 + 2√6) = √3 + √2. Also (√3 − √2)^2 = 3 + 2 − 2√6 = 5 − 2√6, so √(5 − 2√6) = √3 − √2 (positive since √3 > √2). Then use the reciprocal of a conjugate: 1/(√3 − √2) = √3 + √2 after rationalizing. Step-by-Step Solution:Let A = √(5 + 2√6) = √3 + √2Let B = √(5 − 2√6) = √3 − √2Compute 1/B = (√3 + √2) since B(√3 + √2) = (3 − 2) = 1Therefore expression = A − 1/B = (√3 + √2) − (√3 + √2) = 0 Verification / Alternative check:Approximate numerically: √(5 + 2√6) ≈ 3.146, √(5 − 2√6) ≈ 0.318; then 1/0.318 ≈ 3.146; the difference is ≈ 0. Why Other Options Are Wrong:2√2, 2√3, and √5 − 1 are distractors that arise if you mishandle conjugates or skip rationalization. Common Pitfalls:Forgetting that 1/(√3 − √2) equals √3 + √2 and not (√3 − √2)/(1). Rationalization is essential. Final Answer:0

Question 5

Solve the exponential sum (variant): If 2^(x−1) + 2^(x+1) = 320, determine x.

Accepted Answer

Introduction / Context:This is the same structure as a previous item with a different right-hand side. We again factor a shared power of 2 to convert the equation to a simple power-of-two equality. Given Data / Assumptions: 2^(x−1) + 2^(x+1) = 320 Standard index laws apply. Concept / Approach:Factor 2^(x−1) and use 2^(x+1) = 2^(x−1)*2^2. Solve the resulting equation by recognizing powers of 2. Step-by-Step Solution:2^(x−1)(1 + 2^2) = 2^(x−1)*5 = 3202^(x−1) = 320/5 = 64 = 2^6x − 1 = 6 ⇒ x = 7 Verification / Alternative check:Check: 2^6 + 2^8 = 64 + 256 = 320, correct. Why Other Options Are Wrong:Any value other than 7 fails to satisfy the exact equality. Common Pitfalls:Arithmetic slip dividing 320 by 5. Ensure 320/5 = 64. Final Answer:7

Question 6

Indices with chained definitions: If a^x = b, b^y = c and x*y*z = 1, find the value of c^z in terms of a, b, c.

Accepted Answer

Introduction / Context:This checks your comfort with composing exponents when variables are defined by earlier exponent relations. The key is to rewrite everything in a single base, then use the product of exponents condition xyz = 1. Given Data / Assumptions: a^x = b b^y = c x*y*z = 1 Concept / Approach:Express b and c in terms of a. Then raise c to the power z and simplify the exponent using xyz = 1. This turns c^z into a familiar power of a. Step-by-Step Solution:b = a^xc = b^y = (a^x)^y = a^(xy)c^z = (a^(xy))^z = a^(xyz)Given xyz = 1 ⇒ c^z = a^1 = a Verification / Alternative check:Pick sample values x = 1, y = 1, z = 1 (so xyz = 1). Then b = a, c = a, and c^z = a^1 = a, as expected. Why Other Options Are Wrong:b or c would require xyz to equal 1/x or 1/(xy); ab or a/b do not follow from the exponent composition. Common Pitfalls:Forgetting that (a^m)^n = a^(mn). Avoid adding exponents here—multiplication is the rule. Final Answer:a

Question 7

Rewrite to powers of 2 and compare exponents: If 16 × 8^(n+2) = 2^m, find m in terms of n.

Accepted Answer

Introduction / Context:Here we convert mixed bases (16 and 8) to a single base 2, then equate exponents. This is a routine indices exercise reinforcing base conversion and exponent addition. Given Data / Assumptions: 16 × 8^(n+2) = 2^m 16 = 2^4 and 8 = 2^3 Concept / Approach:Rewrite everything as powers of 2. Add exponents on the left (since same base multiplied), then set the combined exponent equal to m. Step-by-Step Solution:16 = 2^48^(n+2) = (2^3)^(n+2) = 2^(3(n+2)) = 2^(3n + 6)LHS = 2^4 × 2^(3n + 6) = 2^(4 + 3n + 6) = 2^(3n + 10)Thus 2^m = 2^(3n + 10) ⇒ m = 3n + 10 Verification / Alternative check:Plug n = 0: LHS = 16 × 8^2 = 16 × 64 = 1024 = 2^10 ⇒ m = 10, consistent with 3(0)+10. Why Other Options Are Wrong:They reflect incorrect coefficient arithmetic (e.g., forgetting the +6 or mis-scaling the 3 on n). Common Pitfalls:Expanding 3(n+2) incorrectly or missing the 2^4 factor from 16. Final Answer:3n + 10

Question 8

Infinite nested radical (repaired): Evaluate S = √(56 + √(56 + √(56 + …))).

Accepted Answer

Introduction / Context:The database text contained a likely typo “÷ 22”. By the Recovery-First Policy, we restore the standard infinite nested radical form S = √(56 + √(56 + …)). Such expressions are solved by setting S equal to the entire radical and squaring to remove the root. Given Data / Assumptions: S = √(56 + S) with S > 0. Convergent positive solution is expected. Concept / Approach:Let S represent the entire expression. Then square both sides to obtain a quadratic in S. Solve and take the positive root because S is a principal square root value. Step-by-Step Solution:S = √(56 + S)Square: S^2 = 56 + SBring terms together: S^2 − S − 56 = 0Solve quadratic: Discriminant D = 1 + 224 = 225, √D = 15S = [1 ± 15]/2 ⇒ S = 8 or S = −7Since S is positive, S = 8 Verification / Alternative check:Plug back: √(56 + 8) = √64 = 8, consistent. Why Other Options Are Wrong:0, 1, 2, 4 do not satisfy S = √(56 + S). Only S = 8 works. Common Pitfalls:Keeping the negative root −7; nested radicals defined via principal roots yield nonnegative values. Final Answer:8

Question 9

Rationalizing trick: If x = (√3 + 1)/(√3 − 1) and y = (√3 − 1)/(√3 + 1), compute x^2 + y^2.

Accepted Answer

Introduction / Context:This tests recognition that x and y are reciprocals when formed as complementary conjugate ratios. From xy = 1, compute x^2 + y^2 efficiently using the identity (x + 1/x)^2 = x^2 + 2 + 1/x^2. Given Data / Assumptions: x = (√3 + 1)/(√3 − 1) y = (√3 − 1)/(√3 + 1) All radicals are principal. Concept / Approach:First, observe x·y = 1 (they are reciprocals). Compute S = x + 1/x directly using conjugates, then obtain x^2 + y^2 = x^2 + 1/x^2 = S^2 − 2. Step-by-Step Solution:Let a = √3, b = 1x = (a + b)/(a − b), 1/x = (a − b)/(a + b)S = x + 1/x = [(a + b)^2 + (a − b)^2]/(a^2 − b^2)= [a^2 + 2ab + b^2 + a^2 − 2ab + b^2]/(a^2 − b^2)= [2a^2 + 2b^2]/(a^2 − b^2) = 2(a^2 + b^2)/(a^2 − b^2)With a^2 = 3, b^2 = 1 ⇒ S = 2(3 + 1)/(3 − 1) = 8/2 = 4Thus x^2 + y^2 = S^2 − 2 = 16 − 2 = 14 Verification / Alternative check:Numerically compute x and y, square, and sum to confirm 14. Why Other Options Are Wrong:They reflect arithmetic slips in evaluating S or in subtracting 2 to obtain x^2 + 1/x^2. Common Pitfalls:Forgetting that x·y = 1, which enables the shortcut; otherwise the computation becomes messy. Final Answer:14

Question 10

Indices clean-up: Find m − n if [(9^n * 3^2 * (3^(−n/2))^(−2) − (27)^n) / (3^(3m) * 2^3)] = 1/27.

Accepted Answer

Introduction / Context:Simplify a complex-looking indices expression by rewriting every term with a base of 3 and factoring out common powers. Careful exponent algebra quickly collapses the fraction. Given Data / Assumptions: (9^n * 3^2 * (3^(−n/2))^(−2) − (27)^n) / (3^(3m) * 2^3) = 1/27 9 = 3^2 and 27 = 3^3 (3^(−n/2))^(−2) = 3^n Concept / Approach:Convert 9^n and 27^n into base 3. Combine exponents in the numerator, factor out the highest common power, and cancel the 2^3 factor with the numerical factor that appears. Then equate powers of 3. Step-by-Step Solution:9^n = (3^2)^n = 3^(2n)(3^(−n/2))^(−2) = 3^nNumerator: 3^(2n) * 3^2 * 3^n − (3^3)^n = 3^(3n+2) − 3^(3n) = 3^(3n)(3^2 − 1) = 3^(3n) * 8Denominator: 3^(3m) * 2^3 = 3^(3m) * 8Fraction simplifies to 3^(3n)/3^(3m) = 3^(3n−3m)Set equal to 1/27 = 3^(−3) ⇒ 3n − 3m = −3 ⇒ n − m = −1 ⇒ m − n = 1 Verification / Alternative check:Pick m = 2, n = 1 (so m − n = 1) to test; the simplified exponent gives −3 as required. Why Other Options Are Wrong:They result from sign errors when moving terms or mishandling the (−n/2)^(−2) step. Common Pitfalls:Dropping the factor (3^2 − 1) = 8 or forgetting to cancel 2^3 in the denominator. Final Answer:1

Question 11

Algebraic division with indices: Find the quotient of (a^(−1) − 1) divided by (a − 1).

Accepted Answer

Introduction / Context:This checks algebraic manipulation with negative exponents and simple factorization. Rewriting a^(−1) as 1/a keeps the work transparent. Given Data / Assumptions: Compute ((a^(−1) − 1) / (a − 1)) for a ≠ 0 and a ≠ 1. Concept / Approach:Rewrite a^(−1) as 1/a, factor out signs to match a − 1, and reduce the fraction carefully to avoid sign errors. Step-by-Step Solution:(a^(−1) − 1)/(a − 1) = ((1/a) − 1)/(a − 1)= ((1 − a)/a)/(a − 1)= ((−(a − 1))/a)/(a − 1)= (−1/a) * ((a − 1)/(a − 1)) = −1/a Verification / Alternative check:Pick a = 2: numerator = 1/2 − 1 = −1/2; denominator = 1; quotient = −1/2 = −1/a. Why Other Options Are Wrong:They come from sign mistakes or inverting the wrong term during simplification. Common Pitfalls:Dropping the minus sign when converting (1 − a) to −(a − 1); ensure domain excludes a = 0, 1. Final Answer:-1/a

Question 12

Absolute difference of small powers: What is the absolute difference between 2^3 and 3^2?

Accepted Answer

Introduction / Context:The original phrasing “difference between” can be ambiguous in sign. By the Recovery-First Policy, we interpret it as absolute difference because MCQ options list positive numbers. We compare 2^3 and 3^2, two familiar small powers. Given Data / Assumptions: 2^3 = 8 3^2 = 9 Report absolute difference |8 − 9|. Concept / Approach:Compute both powers exactly and subtract. Taking the absolute value avoids sign confusion and matches the positive options provided. Step-by-Step Solution:2^3 = 83^2 = 9Absolute difference = |8 − 9| = 1 Verification / Alternative check:Reverse order: |9 − 8| = 1 gives the same result, confirming independence from order. Why Other Options Are Wrong:2, 3, 5, 0 do not equal the magnitude of the difference here. Common Pitfalls:Reporting −1 instead of 1; in word problems “difference” often implies magnitude unless directed otherwise. Final Answer:1

Question 13

Combine like bases (indices law): Evaluate a^5 × a^7 and express the result as a single power of a.

Accepted Answer

Introduction / Context:This is a direct application of the index law a^m * a^n = a^(m+n). We combine exponents with the same base by simple addition. Given Data / Assumptions: a^5 × a^7 a ≠ 0 (usual nonzero base assumption) Concept / Approach:When multiplying powers with the same base, add exponents: m + n. No further simplification is needed beyond adding 5 and 7. Step-by-Step Solution:a^5 × a^7 = a^(5 + 7) = a^12 Verification / Alternative check:For a = 2: 2^5 × 2^7 = 32 × 128 = 4096 = 2^12, as expected. Why Other Options Are Wrong:a^35 multiplies exponents incorrectly; a^2 subtracts instead of adds; a^(5/7) is unrelated; a^10 is adding wrongly. Common Pitfalls:Multiplying exponents (5*7) instead of adding—only (a^m)^n multiplies exponents. Final Answer:a^12

Question 14

Evaluate the cube (third power): Compute 4^3 exactly.

Accepted Answer

Introduction / Context:Simple powers are foundational: 4^3 means 4 multiplied by itself three times. This reinforces exponent interpretation as repeated multiplication. Given Data / Assumptions: Evaluate 4^3. Concept / Approach:Use repeated multiplication: 4^3 = 4 * 4 * 4. Step-by-Step Solution:4 * 4 = 1616 * 4 = 64 Verification / Alternative check:Recognize 2^6 = 64; since 4 = 2^2, (2^2)^3 = 2^6 = 64, consistent. Why Other Options Are Wrong:81 is 3^4 or 9^2; 12 is 3*4; 49 is 7^2; 16 is 4^2, not 4^3. Common Pitfalls:Confusing squaring with cubing; ensure three factors of 4. Final Answer:64

Question 15

Negative and fractional exponents: Simplify (x^(2/3))^(−3/4) and express it with positive exponents if possible.

Accepted Answer

Introduction / Context:This checks the power-of-a-power rule and handling negative exponents. We convert the nested exponent to a single exponent on x, then rewrite with a positive exponent as a reciprocal if needed. Given Data / Assumptions: Expression: (x^(2/3))^(−3/4) x > 0 for real radical interpretation. Concept / Approach:Use (a^m)^n = a^(mn). Multiply exponents 2/3 and −3/4 to get a single exponent. A negative exponent −k means 1/x^k. Convert x^(−1/2) to 1/√x. Step-by-Step Solution:(x^(2/3))^(−3/4) = x^((2/3)*(−3/4)) = x^(−6/12) = x^(−1/2)x^(−1/2) = 1 / x^(1/2) = 1/√x Verification / Alternative check:Test x = 16: LHS = (16^(2/3))^(−3/4). 16^(2/3) = (2^4)^(2/3) = 2^(8/3). Raising to −3/4 yields 2^(−2) = 1/4. RHS 1/√16 = 1/4—matches. Why Other Options Are Wrong:1/x and 1/x^2 correspond to exponents −1 and −2; 1/x^(−2) simplifies to x^2; √x is the reciprocal of the answer. Common Pitfalls:Adding instead of multiplying exponents inside a power-of-a-power; mishandling negative signs. Final Answer:1/√x

Question 16

Combine exponents with same base: Solve 17^(3.5) × 17^(7.3) ÷ 17^(4.2) = 17^? and find the value of ?.

Accepted Answer

Introduction / Context:With the same base, multiply by adding exponents and divide by subtracting exponents. This item uses decimal exponents to ensure comfort beyond integers. Given Data / Assumptions: Base 17 throughout Expression: 17^(3.5 + 7.3 − 4.2) Concept / Approach:a^m * a^n = a^(m+n); a^m / a^n = a^(m−n). Apply sequentially: first add 3.5 and 7.3, then subtract 4.2. Step-by-Step Solution:3.5 + 7.3 = 10.810.8 − 4.2 = 6.6Therefore the expression equals 17^(6.6) Verification / Alternative check:Rearrange: (3.5 − 4.2) + 7.3 = (−0.7) + 7.3 = 6.6—same result. Why Other Options Are Wrong:8, 8.4, 6.4, 7.2 stem from arithmetic slips when combining decimals. Common Pitfalls:Adding all three numbers or misplacing the subtraction; track signs carefully. Final Answer:6.6

Question 17

Solve for the exponent: If 289 = 17^(x/5), find the value of x.

Accepted Answer

Introduction / Context:Recognize perfect powers: 289 is 17 squared. Equating exponents on the same base determines x straightforwardly. Given Data / Assumptions: 289 = 17^(x/5) 17^2 = 289 Concept / Approach:Rewrite 289 as 17^2. Then match exponents: x/5 = 2, solve for x by multiplying both sides by 5. Step-by-Step Solution:17^(x/5) = 17^2 ⇒ x/5 = 2x = 2 * 5 = 10 Verification / Alternative check:Substitute back: 17^(10/5) = 17^2 = 289, correct. Why Other Options Are Wrong:They result from mis-scaling the fraction or confusing 289 with another square. Common Pitfalls:Forgetting to multiply by 5; ensure exact base matching before equating exponents. Final Answer:10

Question 18

Definition check (surd order): L√M denotes the L-th root of M. If M is rational, L is a positive integer, and L√M is irrational, then the surd is of what order?

Accepted Answer

Introduction / Context:A surd is an irrational root expression. The “order” refers to the index of the radical (the number indicating which root). For L√M, L is the order (e.g., cube root has order 3). Given Data / Assumptions: Expression form: L√M, i.e., the L-th root of M. M is rational; L ∈ positive integers. L√M is irrational (hence a surd). Concept / Approach:By definition, in n√A the order (or index) is n. The irrationality condition just certifies it qualifies as a surd; it does not change the order. Step-by-Step Solution:Identify index: LTherefore, the surd is of order L Verification / Alternative check:Examples: √2 (order 2), 3√5 (order 3), 5√7 (order 5). The order equals the radical’s index. Why Other Options Are Wrong:M is the radicand, not the order; 2 or 4 are specific indices, not general; L/2 is unrelated. Common Pitfalls:Confusing the radicand (inside the root) with the index (outside, as the small number on the radical sign). Final Answer:L

Question 19

Exponential equations system (repaired exponents): If 2^p + 3^q = 17 and 2^(p+2) − 3^(q+1) = 5, find ordered pair (p, q).

Accepted Answer

Introduction / Context:The database text omits exponent markers on the second equation. Using the Recovery-First Policy, we repair it to a standard pair of exponential equations. Substitution via linearization trick (set A = 2^p, B = 3^q) reduces the system to linear equations in A and B. Given Data / Assumptions: 2^p + 3^q = 17 2^(p+2) − 3^(q+1) = 5 2^(p+2) = 4·2^p and 3^(q+1) = 3·3^q Concept / Approach:Let A = 2^p and B = 3^q. Then equations become A + B = 17 and 4A − 3B = 5. Solve the linear system for A and B, then back-solve for p and q by recognizing small powers of 2 and 3. Step-by-Step Solution:A + B = 17 … (1)4A − 3B = 5 … (2)From (1): B = 17 − APlug in (2): 4A − 3(17 − A) = 5 ⇒ 4A − 51 + 3A = 5 ⇒ 7A = 56 ⇒ A = 8Thus 2^p = 8 ⇒ p = 3; and B = 17 − 8 = 9 = 3^2 ⇒ q = 2 Verification / Alternative check:Check both equations: 2^3 + 3^2 = 8 + 9 = 17; 2^5 − 3^3 = 32 − 27 = 5, both true. Why Other Options Are Wrong:They do not solve the linear system correctly or do not match integer powers of 2 and 3. Common Pitfalls:Forgetting 2^(p+2) = 4·2^p and 3^(q+1) = 3·3^q; mixing addition and subtraction signs. Final Answer:3, 2

Question 20

Evaluate the expression carefully using index rules and cube roots: (42 × 229) ÷ (9261)^(1/3) = ?

Accepted Answer

Introduction / Context: This problem checks your comfort with order of operations and perfect cubes. Recognizing that 9261 is a perfect cube allows a short and clean computation without a calculator. Once the cube root is simplified, the expression reduces to a straightforward integer division after multiplying two integers. Given Data / Assumptions: Expression: (42 × 229) ÷ (9261)^(1/3). Standard order of operations: evaluate the root first, then perform multiplication and division left to right. All numbers are integers; 9261 is suspected to be a perfect cube. Concept / Approach: Use the identity that if N = a^3, then N^(1/3) = a. Here, 21^3 = 9261. So (9261)^(1/3) = 21. Then multiply 42 by 229 and divide the product by 21. Because 42 is divisible by 21, we can simplify before multiplying to avoid large intermediate numbers. Step-by-Step Solution: Compute the cube root: (9261)^(1/3) = 21.Rewrite the expression: (42 × 229) ÷ 21.Simplify first: 42 ÷ 21 = 2.Now multiply: 2 × 229 = 458. Verification / Alternative check: Multiply first then divide: 42 × 229 = 9618; 9618 ÷ 21 = 458. Both paths agree, confirming the result. Why Other Options Are Wrong: 448, 452, 456, and 462 are near the true value but occur if you mis-evaluate the cube root or make a small multiplication/division slip. Common Pitfalls: Treating 9261^(1/3) as 31 or 19 by guesswork, or dividing 229 by 21 instead of simplifying 42 ÷ 21 first. Always simplify where possible to reduce arithmetic errors. Final Answer: 458

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