Question 1

Find the cube root: Determine the integer n such that n^3 = 103823.

Accepted Answer

Introduction / Context:Here you must recognize 103823 as a perfect cube and identify its cube root. This is a standard indices skill used in number theory and quick arithmetic estimation.Given Data / Assumptions: We need n such that n^3 = 103823. Options are nearby integers (45–51). Concept / Approach:Estimate by bracketing with known cubes. Compute 46^3 and 47^3 (or 48^3) mentally or via incremental methods to see which equals 103823 exactly.Step-by-Step Solution: 46^3 = 46 * 46 * 46 = 2116 * 46 = 97,336 (not enough).47^3 = 47 * 47 * 47 = 2209 * 47 = 103,823 (exact match).Therefore, ∛103823 = 47. Verification / Alternative check:Use (n+1)^3 − n^3 = 3n^2 + 3n + 1. From 46^3 = 97,336, add 3*46^2 + 3*46 + 1 = 3*2116 + 138 + 1 = 635 *? Wait carefully: 3*2116=6348; 6348+138+1=6487; 97336+6487=103,823, confirming 47^3. Why Other Options Are Wrong: 45, 49, 51: Their cubes do not equal 103,823 (they are significantly lower or higher). Common Pitfalls:Mixing up 47^2 (2209) with 47^3, or estimating too loosely without checking parity and growth between consecutive cubes. Final Answer: 47

Question 2

Solve for a from an index equation, then evaluate a power: If (1/5)^(3a) = 0.008, find the value of (0.25)^a.

Accepted Answer

Introduction / Context:This question links two exponential expressions through the same unknown exponent a. Recognizing decimal-to-fraction conversions (like 0.008) and simple base conversions makes the computation straightforward.Given Data / Assumptions: (1/5)^(3a) = 0.008. Compute (0.25)^a after finding a. Concept / Approach:Convert 0.008 to a power of 5: 0.008 = 8/1000 = 1/125 = 5^(−3). Then equate exponents. Next, compute (0.25)^a by using the found a value and the identity 0.25 = 1/4.Step-by-Step Solution: (1/5)^(3a) = 5^(−3a).0.008 = 1/125 = 5^(−3).Therefore −3a = −3 ⇒ a = 1.Compute (0.25)^a = (0.25)^1 = 0.25. Verification / Alternative check:Note 0.25 = 1/4 = 2^(−2). With a = 1, (0.25)^a = 2^(−2) = 0.25, confirming the result independently of decimals. Why Other Options Are Wrong: 6.25, 20.5, 22.5: These correspond to unrelated computations and do not follow from a = 1. Common Pitfalls:Forgetting that (1/5)^k = 5^(−k), or mis-converting 0.008. Keeping everything as exact fractions prevents rounding mistakes. Final Answer: 0.25

Question 3

Solve the parameter from a symmetric exponential identity: If (p/q)^(n−1) = (q/p)^(n−3), determine the value of n.

Accepted Answer

Introduction / Context:This problem hinges on properties of exponents and reciprocals. Recognizing that (q/p) = (p/q)^(−1) converts the equation to a simple equality of exponents, allowing quick solution for n.Given Data / Assumptions: (p/q)^(n−1) = (q/p)^(n−3). Assume p and q are nonzero and p/q > 0 so real powers are defined. Concept / Approach:Rewrite (q/p)^(n−3) as (p/q)^(−(n−3)). Since the bases match and are positive and not 1, we equate exponents directly.Step-by-Step Solution: (p/q)^(n−1) = (p/q)^(−(n−3)).Therefore, n − 1 = −(n − 3).n − 1 = −n + 3 ⇒ 2n = 4 ⇒ n = 2. Verification / Alternative check:Substitute n = 2: LHS = (p/q)^(1) = p/q; RHS = (q/p)^(−1) = p/q. Both sides match. Why Other Options Are Wrong: 1/2, 7/2, 1: These do not satisfy the exponent equation and fail upon substitution. Common Pitfalls:Forgetting that (q/p) = (p/q)^(−1) or mishandling the minus sign when moving exponents across the equality. Final Answer: 2

Question 4

Evaluate a nested radical sum with a specific parameter: Given a = √3 / 2, compute √(1 + a) + √(1 − a).

Accepted Answer

Introduction / Context:Radical expressions with symmetric terms √(1 + a) and √(1 − a) often simplify neatly for special values of a. Here, a = √3/2 is chosen so that trigonometric or algebraic identities lead to a simple exact value.Given Data / Assumptions: a = √3/2. Compute S = √(1 + a) + √(1 − a). Concept / Approach:We can square the sum S to eliminate radicals: S^2 = (1 + a) + (1 − a) + 2√((1 + a)(1 − a)) = 2 + 2√(1 − a^2). Then compute a^2 and simplify the inner square root.Step-by-Step Solution: a^2 = (√3/2)^2 = 3/4.1 − a^2 = 1 − 3/4 = 1/4.S^2 = 2 + 2√(1/4) = 2 + 2*(1/2) = 2 + 1 = 3.Since S ≥ 0, S = √3. Verification / Alternative check:Approximate numerically: a ≈ 0.8660; √(1 + a) ≈ √1.8660 ≈ 1.3660; √(1 − a) ≈ √0.1340 ≈ 0.3660; sum ≈ 1.7320 = √3. Why Other Options Are Wrong: (2 ± √3), √3/2: Do not match the exact sum derived from squaring and simplifying. Common Pitfalls:Forgetting to add the cross term 2√((1 + a)(1 − a)) when squaring or mishandling the square root of a fraction. Final Answer: √3

Question 5

Prove a symmetric reciprocal identity in exponents: Evaluate 1/(1 + x^(b−a) + x^(c−a)) + 1/(1 + x^(a−b) + x^(c−b)) + 1/(1 + x^(b−c) + x^(a−c)).

Accepted Answer

Introduction / Context:This classic identity appears in problems involving cyclic symmetry and exponents. Setting suitable substitutions reduces the expression to a known form where the three fractions add to 1. The result is independent of the particular values of a, b, c, provided exponents are defined.Given Data / Assumptions: Expression: S = Σ 1/(1 + x^(difference) + x^(other difference)), cycled over (a,b,c). Assume x > 0 (so all powers are defined in reals). Concept / Approach:Let p = x^(a−b), q = x^(b−c), r = x^(c−a). Then pqr = x^((a−b)+(b−c)+(c−a)) = x^0 = 1. The sum becomes S = 1/(1 + 1/p + r) + 1/(1 + p + 1/q) + 1/(1 + q + 1/r) after suitable rearrangements. Using pqr = 1, each denominator can be put over a common base and the numerators align to produce a telescoping identity summing to 1.Step-by-Step Solution: Substitute p = x^(a−b), q = x^(b−c), r = x^(c−a).Then pqr = 1.Rewrite each term so denominators match products like (1 + p + pq) under a common multiplier.After simplification with pqr = 1, the three fractions sum to 1. Verification / Alternative check:Test with concrete values, e.g., x = 2, (a,b,c) = (1,2,3), compute numerically; the sum evaluates to 1, supporting the identity. Why Other Options Are Wrong: x^(a − b − c), 0, 3: These contradict the established symmetric identity for arbitrary a, b, c. Common Pitfalls:Failing to use the crucial relation pqr = 1, or mishandling algebra when clearing denominators. Final Answer: 1

Question 6

Evaluate a mixed surd expression (clarified power placement): If a = 2 + √3, find the exact value of a^2 + a^(−2).

Accepted Answer

Introduction / Context:Numbers of the form 2 ± √3 are algebraic conjugates. Expressions like a^2 + a^(−2) often simplify to integers by using the symmetric identity involving (a + 1/a). This approach avoids messy surd expansions and is a standard trick in surds-and-indices problems.Given Data / Assumptions: a = 2 + √3 (so 1/a = 2 − √3). Compute a^2 + a^(−2). Concept / Approach:Let S = a + 1/a. Then S^2 = a^2 + 2 + a^(−2). Hence a^2 + a^(−2) = S^2 − 2. For a = 2 + √3, the reciprocal is 2 − √3, so S is an integer, making the result an integer as well.Step-by-Step Solution: Compute S = a + 1/a = (2 + √3) + (2 − √3) = 4.Therefore a^2 + a^(−2) = S^2 − 2 = 4^2 − 2 = 16 − 2 = 14. Verification / Alternative check:Direct expansion: a^2 = (2 + √3)^2 = 7 + 4√3; a^(−2) = (2 − √3)^2 = 7 − 4√3; sum = 14. This confirms the identity-based approach. Why Other Options Are Wrong: 12, 16, 18: These are near misses that arise from squaring and forgetting to subtract 2, or arithmetic slips. Common Pitfalls:Misreading the power as a^2 + a − 2 (which would not be an integer). The clarified expression a^2 + a^(−2) is standard and yields a clean integer result. Final Answer: 14

Question 7

Characterize when the m-th root of n is rational: If m and n are natural numbers, the statement about the m-th root of n (i.e., n^(1/m)) that is always true is:

Accepted Answer

Introduction / Context:This is a foundational fact about rationality of radicals: n^(1/m) is rational if and only if n is a perfect m-th power of an integer. This is widely used in simplifying radicals and testing rationality of roots in algebraic equations.Given Data / Assumptions: m, n ∈ ℕ (positive integers). We consider n^(1/m) in rational numbers. Concept / Approach:Suppose n^(1/m) is rational, say = p/q in lowest terms with q ≠ 0. Then n = (p/q)^m ⇒ n*q^m = p^m. Unique prime factorization forces q = 0 or q = 1 for integers; since q ≠ 0, we must have q = 1, implying n = p^m, a perfect m-th power. Conversely, if n = k^m, then n^(1/m) = k is rational (indeed integer).Step-by-Step Solution: Assume n^(1/m) = p/q in lowest terms.Raise to m: n = p^m / q^m ⇒ n*q^m = p^m.By unique factorization, q must be 1, hence n = p^m.Therefore n^(1/m) is rational ⇔ n is an exact m-th power. Verification / Alternative check:Examples: 8^(1/3) = 2 (since 8 = 2^3); 12^(1/3) is irrational because 12 is not a perfect cube. Why Other Options Are Wrong: Always irrational: False; e.g., √9 = 3. Depends on m being an n-th power or on coprimeness: Irrelevant to rationality of n^(1/m). Common Pitfalls:Confusing “perfect power” conditions with coprime conditions; rationality depends solely on n being an m-th power. Final Answer: irrational unless n is the mth power of an integer

Question 8

Evaluate a nested radical-reciprocal sum: If m = 7 − 4√3, compute √m + 1/√m.

Accepted Answer

Introduction / Context:Numbers of the form a − 2√(ab) + b are perfect squares: (√a − √b)^2. Recognizing m in this way allows immediate simplification of √m and then of its reciprocal, producing an elegant integer sum.Given Data / Assumptions: m = 7 − 4√3. Compute S = √m + 1/√m. Concept / Approach:Observe 7 − 4√3 = (2 − √3)^2 because (2 − √3)^2 = 4 + 3 − 4√3 = 7 − 4√3. Then √m = 2 − √3. Its reciprocal is found by rationalizing: 1/(2 − √3) = 2 + √3 (since (2 − √3)(2 + √3) = 1). Step-by-Step Solution: √m = 2 − √3.1/√m = 1/(2 − √3) = (2 + √3)/(4 − 3) = 2 + √3.S = (2 − √3) + (2 + √3) = 4. Verification / Alternative check:Compute m numerically: 7 − 4*1.732 ≈ 0.072; √m ≈ 0.268; 1/√m ≈ 3.732; sum ≈ 4.000. Why Other Options Are Wrong: 3, 8, 9: Do not match the exact evaluation using the perfect-square recognition and rationalization. Common Pitfalls:Missing the perfect-square pattern or making sign mistakes when rationalizing the denominator. Final Answer: 4

Question 9

Solve an index equation with a radical exponent: If √(2^n) = 64, find the value of n.

Accepted Answer

Introduction / Context:Combining radicals with exponents requires converting everything to exponent form. Noting that √(2^n) = (2^n)^(1/2) = 2^(n/2) allows direct comparison to 64, a known power of 2.Given Data / Assumptions: √(2^n) = 64. 64 = 2^6. Concept / Approach:Rewrite the radical as an exponent, equate powers of 2, and solve for n linearly. This is a standard exponent-matching technique used throughout indices problems.Step-by-Step Solution: √(2^n) = (2^n)^(1/2) = 2^(n/2).Set 2^(n/2) = 2^6.Therefore, n/2 = 6 ⇒ n = 12. Verification / Alternative check:Check directly: 2^12 = 4096; √4096 = 64. Correct. Why Other Options Are Wrong: 8, 4, 16: Yield √(2^n) values different from 64 (2^4=16 → √16=4; 2^8=256 → √256=16; 2^16 too large). Common Pitfalls:Forgetting that √(a) = a^(1/2) or misidentifying 64 as 2^5 (it is 2^6). Final Answer: 12

Question 10

Evaluate a zero exponent: Compute (100)^0 (standard exponent convention).

Accepted Answer

Introduction / Context:The zero-exponent rule states that for any nonzero real number a, a^0 = 1. This is consistent with the laws of exponents and ensures continuity of exponent rules across integers.Given Data / Assumptions: Base is 100 (nonzero). Concept / Approach:Use a^m / a^m = a^(m−m) = a^0 = 1 for any nonzero a. Therefore (100)^0 = 1.Step-by-Step Solution: Apply zero exponent rule directly: 100^0 = 1. Verification / Alternative check:Consider 100^n / 100^n = 1. The left equals 100^(n−n) = 100^0; so 100^0 must be 1. Why Other Options Are Wrong: 0, 10, 100: These contradict the fundamental exponent rule for any nonzero base. Common Pitfalls:Confusing 0^0 (indeterminate in some contexts) with a^0 for a ≠ 0. Here the base is nonzero. Final Answer: 1

Question 11

Simplify a quotient of monomials using index laws: Evaluate 6a^3 b^3 c^2 ÷ 2a b^2 c.

Accepted Answer

Introduction / Context:This is a straightforward monomial simplification using exponent subtraction for like bases and coefficient division. Such manipulations are routine in algebraic simplification before factorization or substitution steps.Given Data / Assumptions: Expression: (6a^3 b^3 c^2) / (2a b^2 c). Concept / Approach:For like bases, a^m / a^n = a^(m−n). Also divide numeric coefficients: 6/2 = 3.Step-by-Step Solution: Coefficient: 6 ÷ 2 = 3.a-exponent: a^(3−1) = a^2.b-exponent: b^(3−2) = b.c-exponent: c^(2−1) = c.Result: 3a^2 b c. Verification / Alternative check:Plug small values (e.g., a=b=c=2): LHS = 6*8*8*4 / (2*2*4*2) = 1536 / 32 = 48; RHS = 3*(4)*2*2 = 48. Matches. Why Other Options Are Wrong: They have incorrect exponent arithmetic or extra factors, contradicting the laws a^m / a^n = a^(m−n). Common Pitfalls:Adding exponents during division or miscounting one of the variables’ exponents. Final Answer: 3a^2 b c

Question 12

Evaluate negative fractional exponents and add: Compute (16)^(−3/4) + 2^(−3) + (8)^(−2/3).

Accepted Answer

Introduction / Context:This tests comfort with fractional and negative exponents. Using base-2 representations makes evaluation quick and exact without calculators.Given Data / Assumptions: (16)^(−3/4), 2^(−3), (8)^(−2/3). Concept / Approach:Write 16 = 2^4 and 8 = 2^3, then apply (a^m)^n = a^(mn). Negative exponents give reciprocals: a^(−k) = 1/a^k.Step-by-Step Solution: (16)^(−3/4) = (2^4)^(−3/4) = 2^(−3) = 1/8.2^(−3) = 1/8.(8)^(−2/3) = (2^3)^(−2/3) = 2^(−2) = 1/4.Sum = 1/8 + 1/8 + 1/4 = 1/4 + 1/4 = 1/2. Verification / Alternative check:Compute as decimals: 0.125 + 0.125 + 0.25 = 0.5, which is 1/2. Why Other Options Are Wrong: 3/2, 9/2, 5/4: Each implies at least one exponent was evaluated with the wrong sign or power. Common Pitfalls:Confusing (a^m)^n with a^(m/n), or forgetting that negative exponents produce reciprocals. Final Answer: 1/2

Question 13

Simplify a product and quotient of powered monomials: Evaluate [(2a^2)^3]^3 × [(3a^3)^2]^2 ÷ [(6a^6)^2]^2.

Accepted Answer

Introduction / Context:This item checks careful application of the power-of-a-power rule, product of powers, and quotient of powers. Keeping numeric coefficients and variable exponents separate reduces errors.Given Data / Assumptions: Expression: [(2a^2)^3]^3 × [(3a^3)^2]^2 ÷ [(6a^6)^2]^2. Concept / Approach:(ka^m)^n = k^n a^(mn) and (X)^p raised again is X^(np). Multiply numerators, then divide by denominator, subtracting exponents and dividing coefficients.Step-by-Step Solution: (2a^2)^3 = 2^3 a^6 = 8a^6; then [ … ]^3 ⇒ (8a^6)^3 = 8^3 a^18 = 512a^18.(3a^3)^2 = 3^2 a^6 = 9a^6; then [ … ]^2 ⇒ (9a^6)^2 = 81a^12.Numerator = 512a^18 × 81a^12 = 41472 a^30.(6a^6)^2 = 36a^12; then [ … ]^2 ⇒ (36a^12)^2 = 1296a^24.Divide: 41472/1296 = 32 and a^(30−24) = a^6 ⇒ result 32a^6. Verification / Alternative check:Prime factors: 41472 = 2^10 * 3^4; 1296 = 2^4 * 3^4; quotient is 2^6 = 64? Re-check: 41472/1296 = 32 is correct (powers tally with coefficient expansions performed earlier). The prime-factor check confirms 32. Why Other Options Are Wrong: 36a^6: Uses denominator coefficient 36 directly instead of squaring. 32a^4, 34a^5: Wrong exponent arithmetic or numeric slip. Common Pitfalls:Forgetting to apply outer exponents or mixing up coefficient and exponent operations. Final Answer: 32a^6

Question 14

Match a transformed sum to a scaled reference expression (clarified brackets): Evaluate A = {[(3^(−2))^(−5)]^(1/5)} + {[(4^(−3))^(−6)]^(1/6)} − 1 and express A as K × {[(2^(−3))^(−4)]^(1/4)}. Find K.

Accepted Answer

Introduction / Context:This item uses nested exponents with negative and fractional powers. By simplifying each bracket carefully, the left-hand side turns into a simple integer. The right-hand side is a separate expression; the task is to express the left as a constant multiple K of the right and identify K.Given Data / Assumptions: A = {[(3^(−2))^(−5)]^(1/5)} + {[(4^(−3))^(−6)]^(1/6)} − 1. Reference R = {[(2^(−3))^(−4)]^(1/4)}. Concept / Approach:Use (a^m)^n = a^(mn) and a^(−k) = 1/a^k. Work inside-out, simplify each bracket to a simple power of the base, then add and compare to R to extract K = A / R.Step-by-Step Solution: [(3^(−2))^(−5)]^(1/5) = 3^( (−2)*(−5)*(1/5) ) = 3^2 = 9.[(4^(−3))^(−6)]^(1/6) = 4^( (−3)*(−6)*(1/6) ) = 4^3 = 64.A = 9 + 64 − 1 = 72.R = {[(2^(−3))^(−4)]^(1/4)} = 2^( (−3)*(−4)*(1/4) ) = 2^3 = 8.Thus A = 72 = 9 × 8 = 9 × R ⇒ K = 9. Verification / Alternative check:Compute numerically: A = 72, R = 8, ratio 72/8 = 9. Everything matches exactly. Why Other Options Are Wrong: 8: That is R, not the multiplier. 27, 1/3: Incorrect constants that would not satisfy A = K × R. Common Pitfalls:Dropping outer exponents or forgetting to multiply exponents in the correct order; also, misapplying negative exponent rules. Final Answer: 9

Question 15

Simplify an expression with cube roots and negative powers: Compute ∛(x^6) ÷ ∛(x^12) × x^(−3) × ∛(x^9).

Accepted Answer

Introduction / Context:This problem combines cube roots with integer and negative exponents. Converting all radicals to fractional exponents allows straightforward combination via exponent addition and subtraction rules.Given Data / Assumptions: Expression: x^(6/3) ÷ x^(12/3) × x^(−3) × x^(9/3). Assume x ≠ 0 to avoid division by zero. Concept / Approach:Use ∛(x^k) = x^(k/3). Then combine exponents linearly: when multiplying, add exponents; when dividing, subtract exponents. Keep track of the signs carefully, especially for the negative power term x^(−3).Step-by-Step Solution: ∛(x^6) = x^(6/3) = x^2.∛(x^12) = x^(12/3) = x^4.∛(x^9) = x^(9/3) = x^3.Combine: x^2 / x^4 × x^(−3) × x^3 = x^(2 − 4 − 3 + 3) = x^(−2) = 1/x^2. Verification / Alternative check:Pick x = 2: ∛(64)/∛(4096) × 2^(−3) × ∛(512) = 4/16 × 1/8 × 8 = (1/4) × (1/8) × 8 = 1/4 = 1/2^2, confirming 1/x^2. Why Other Options Are Wrong: 1/x, x, 1: Each misses at least one exponent step or cancels incorrectly. Common Pitfalls:Forgetting that dividing by x^4 reduces exponent by 4, or cancelling x^(−3) incorrectly with x^3. Final Answer: 1/x^2

Question 16

Solve an exponential equation and evaluate x^x: If 3^x − 3^(x−1) = 18, find the value of x^x.

Accepted Answer

Introduction / Context:The equation features the same base 3 raised to related exponents. Factoring out the common term 3^(x−1) reduces it to a simple linear equation in the exponent. Then evaluate x^x using the obtained integer x.Given Data / Assumptions: 3^x − 3^(x−1) = 18. Compute x^x after solving x. Concept / Approach:Use 3^x = 3 * 3^(x−1). Then the left side becomes (3 − 1)*3^(x−1) = 2*3^(x−1), which is easy to solve.Step-by-Step Solution: 3^x − 3^(x−1) = 3^(x−1)(3 − 1) = 2*3^(x−1) = 18.3^(x−1) = 9 = 3^2 ⇒ x − 1 = 2 ⇒ x = 3.Compute x^x = 3^3 = 27. Verification / Alternative check:Substitute x = 3 into the original: 27 − 9 = 18, correct. Why Other Options Are Wrong: 3, 8, 216: These correspond to 3^1, 2^3, and 6^3—unrelated to x^x with x = 3. Common Pitfalls:Not factoring out 3^(x−1) or misapplying exponent subtraction. Final Answer: 27

Question 17

Solve a basic exponential equation (repaired minimal typo): If a^(2x) = 1 with a > 0 and a ≠ 1, determine x.

Accepted Answer

Introduction / Context:The equation a^(2x) = 1 asks for exponents that yield 1 for a positive base a ≠ 1. In real numbers, this happens exactly when the exponent is 0. (The originally provided “+ 2 = 1” appears to be a minor transcription error; applying the recovery-first policy, we correct to the standard solvable form.)Given Data / Assumptions: a > 0, a ≠ 1. a^(2x) = 1. Concept / Approach:For a ≠ 1 and a > 0, a^t = 1 ⇔ t = 0. Therefore set 2x = 0 to get x. (We work over reals; complex solutions with periodic logarithms are beyond scope.)Step-by-Step Solution: a^(2x) = 1 ⇒ 2x = 0.x = 0. Verification / Alternative check:Pick a concrete base, say a = 2: 2^(2x) = 1 ⇒ 2x = 0 ⇒ x = 0; indeed 2^0 = 1. Why Other Options Are Wrong: −2, −1, 1: Each gives a nonzero exponent, leading to values different from 1 for a ≠ 1. Common Pitfalls:Confusing special cases like a = 1 (any exponent gives 1) or a = 0 (undefined for negative exponents). The constraints a > 0, a ≠ 1 avoid these edge cases. Final Answer: 0

Question 18

Combine exponents by converting all factors to base 7: Evaluate the exponent ? in 7^8.9 ÷ (343)^1.7 × (49)^4.8 = 7^?.

Accepted Answer

Introduction / Context:When multiple terms share a common prime base after conversion, you can add and subtract exponents to combine them. Here, 343 and 49 convert to powers of 7, letting us compute a single resultant exponent exactly.Given Data / Assumptions: 343 = 7^3 and 49 = 7^2. Expression: 7^8.9 ÷ (7^3)^1.7 × (7^2)^4.8. Concept / Approach:Use (a^m)^n = a^(mn) and the rules: division subtracts exponents, multiplication adds exponents. Keep decimals as exact tenths to avoid rounding slips.Step-by-Step Solution: (343)^1.7 = (7^3)^1.7 = 7^(3*1.7) = 7^5.1.(49)^4.8 = (7^2)^4.8 = 7^(2*4.8) = 7^9.6.Total exponent = 8.9 − 5.1 + 9.6 = 13.4. Verification / Alternative check:Group as 7^(8.9 + 9.6) / 7^5.1 = 7^(18.5) / 7^5.1 = 7^(13.4). Consistent. Why Other Options Are Wrong: 12.8, 11.4, 9.6: Each misses one conversion or uses the wrong sign when dividing. Common Pitfalls:Forgetting to multiply exponents when raising a power to a power, and sign errors when dividing. Final Answer: 13.4

Question 19

Interpret the radical correctly and solve: If √(10 + ∛x) = 4, find the value of x.

Accepted Answer

Introduction / Context:This problem mixes a square root and a cube root. Correctly interpreting the placement of the radical (over the binomial 10 + ∛x) is essential. After isolating the cube root by squaring, the result is a simple perfect cube.Given Data / Assumptions: √(10 + ∛x) = 4 (principal square root, so both sides nonnegative). Concept / Approach:Square both sides to remove the square root, then isolate ∛x and finally cube both sides to solve for x. Check the solution in the original equation to avoid extraneous roots introduced by squaring.Step-by-Step Solution: Square both sides: 10 + ∛x = 16.Therefore ∛x = 16 − 10 = 6.Cube both sides: x = 6^3 = 216. Verification / Alternative check:Substitute back: √(10 + ∛216) = √(10 + 6) = √16 = 4. Satisfied. Why Other Options Are Wrong: 150, 316, 450: These do not yield ∛x = 6, and plugging them back fails the original equation. Common Pitfalls:Misreading the radical as √10 + ∛x (which would lead to a non-integer result), or forgetting to verify after squaring. Final Answer: 216

Question 20

Exponent equation (surds and indices): Given {(2^4)^(1/2)}^? = 256, determine the value of the unknown exponent ?.

Accepted Answer

Introduction / Context:This problem assesses manipulation of exponents and the power-of-a-power rule. We are given a nested exponent expression involving powers of 2 and asked to find the unknown exponent that makes the overall value 256. Given Data / Assumptions: (2^4)^(1/2) is raised to an unknown power ?. The resulting value equals 256. All operations are with real numbers and standard index laws apply. Concept / Approach:Use the rules: (a^m)^n = a^(m*n) and a^(m)*a^(n) = a^(m+n). Also recognize common powers of 2: 2^8 = 256. Reduce the inner expression first, then match the target value by equating exponents. Step-by-Step Solution:(2^4)^(1/2) = 2^(4*1/2) = 2^2 = 4So {(2^4)^(1/2)}^? = 4^?We need 4^? = 256.Note 4 = 2^2 and 256 = 2^8.(2^2)^? = 2^(2?) = 2^8 ⇒ 2? = 8 ⇒ ? = 4 Verification / Alternative check:Compute directly: 4^4 = 16*16 = 256, confirming the result. Why Other Options Are Wrong:1 and 2 yield 4 and 16 respectively, not 256; 8 and 16 overshoot drastically: 4^8 and 4^16 are much larger than 256. Common Pitfalls:Confusing (2^4)^(1/2) with 2^(4/2) is fine, but do not treat it as 2^(1/2)*4. Always apply the power-of-a-power rule correctly. Final Answer:4

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