Chapter SH, Problem 7.16EP

To determine

To simplify

  ${\left(\frac{125}{343}\right)}^{\frac{4}{3}}$ .

Answer to Problem 7.16EP

Simplified form is $\frac{625}{{2401}^{}}$ .

Explanation of Solution

Given information:

Expression

  ${\left(\frac{125}{343}\right)}^{\frac{4}{3}}$

Calculation:

  $={\left(\frac{125}{343}\right)}^{\frac{4}{3}}$

Put $125=5^3$ and $343=7^3$ in above expression

  $={\left(\frac{5^3}{7^3}\right)}^{\frac{4}{3}}$

So,

  $={\left({\left(\frac{5}{7}\right)}^3\right)}^{\frac{4}{3}}$

Using identity

  ${(a^m)}^n=a^{mn}$

Where m and n are natural numbers.

Then,

  $={\left(\frac{5}{7}\right)}^{3\times \frac{4}{3}}$

On solving,

  $={\left(\frac{5}{7}\right)}^4$

On solving,

  $=\frac{5^4}{7^4}$

On solving,

  $=\frac{5\times 5\times 5\times 5}{7\times 7\times 7\times 7^{}}$

On solving,

  $=\frac{625}{{2401}^{}}$

Hence, simplified form is $\frac{625}{{2401}^{}}$ .