Chapter SH, Problem 7.15EP

To determine

To simplify ${4096}^{\frac{1}{3}}$ .

Answer to Problem 7.15EP

Simplified form is 16.

Explanation of Solution

Given information:

Expression

  ${4096}^{\frac{1}{3}}$

Calculation:

  ${4096}^{\frac{1}{3}}$

Rewrite, above $4096={(64)}^2$

  $={\left({(64)}^2\right)}^{\frac{1}{3}}$

Put $64=8^2$ in above

Then,

  $={\left({({(8)}^2)}^2\right)}^{\frac{1}{3}}$

Put $8=2^3$ in above

  $={\left({({({(2)}^3)}^2)}^2\right)}^{\frac{1}{3}}$

Using identity

  ${(a^m)}^n=a^{mn}$

Where m and n are natural numbers.

Then,

  $=2^{3\times 2\times 2\times \frac{1}{3}}$

So,

  $=2^4$

On solving,

  $=16$

Hence, simplified form is $16$ .