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Evaluate the given expression. (641)(−1/3)
2
4⁻¹
2⁻¹
4
4
Express 64 as 43. Since the power is negative, the reciprocal flips the fraction, and the cube root cancels the power of 3: (1/43)−1/3=(43)1/3=41=4.
Expression to evaluate: (1/64)^(-1/3)
(am)n=am×n and x−n=xn1
Express 64 as 43. Since the power is negative, the reciprocal flips the fraction, and the cube root cancels the power of 3: (1/43)−1/3=(43)1/3=41=4.
Students often forget the negative sign on the exponent, calculating (1/64)1/3=1/4 instead of flipping the fraction to get 4.
Rewrite the fraction
We know that 64=43. Thus, the fraction 641 can be written as (43)−1=4−3.
641=4−3
Apply exponent power rule
Substitute 4−3 into the original expression and apply the rule (am)n=am×n.
(4−3)−1/3=4−3×−1/3
Simplify and evaluate
Multiply the exponents: −3×−1/3=1. Therefore, 41=4.
41=4
D is correct because (1/64)^(-1/3) simplifies to (4³)^(1/3) which equals 4.
This type of simplification is essential for solving logarithmic equations and complex surds in algebra.
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