Mathematics

Mathematics

Problem

Simplify 64a6b33(2a−1b12)2\dfrac{\sqrt[3]{64a^{6}b^{3}}}{\bigl(2a^{-1}b^{\tfrac12}\bigr)^{2}}, where a,b>0a,b>0.
Intuitive explanation

The key is to rewrite everything using exponent rules, then simplify systematically. Start by converting the cube root to a fractional exponent and expanding the denominator using the power rule (xy)n=xnyn(xy)^n = x^n y^n. Once both numerator and denominator are written as products of powers, you can cancel matching bases and combine exponents using the quotient rule xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.

Intuitive explanation

The key is to rewrite everything using exponent rules, then simplify systematically. Start by converting the cube root to a fractional exponent and expanding the denominator using the power rule (xy)n=xnyn(xy)^n = x^n y^n. Once both numerator and denominator are written as products of powers, you can cancel matching bases and combine exponents using the quotient rule xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.

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