Prove the following identity for any positive real number : [3] a , [6] a = a^ 1 2

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Prove the following identity for any positive real number aa: a3 a6=a12.\sqrt[3]{a}\,\sqrt[6]{a} = a^{\frac{1}{2}}. Then use the identity to evaluate the product when a=64a = 64. (i) Express both radicals on the left-hand side in exponential form. (ii) Using the laws of exponents, simplify the product to obtain the right-hand side of the identity. (iii) Substitute a=64a = 64 and compute the numerical value of each side to verify the identity.

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