Chapter 7.9, Problem 65WE

To determine

To calculate: The simplified form of the expression $\frac{{\left(3125\right)}^2}{15625}$ .

Answer to Problem 65WE

The simplified form of the expression $\frac{{\left(3125\right)}^2}{15625}$ is $625$ .

Explanation of Solution

Given information:

The expression $\frac{{\left(3125\right)}^2}{15625}$ .

Formula used:

Law of exponent for quotient of powers is $a^m÷a^n=a^{m-n}$ where a is a nonzero integers and n and m are positive integers.

Law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Calculation:

Consider the expression $\frac{{\left(3125\right)}^2}{15625}$ .

Rewrite the above expression with the help of table of 5.

  $\frac{{\left(3125\right)}^2}{15625}=\frac{{\left(5^5\right)}^2}{5^6}$

Recall that law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}\frac{{\left(3125\right)}^2}{15625}=\frac{{\left(5^5\right)}^2}{5^6}\\ =\frac{5^{10}}{5^6}\end{array}$

Recall that law of exponent for quotient of powers is $a^m÷a^n=a^{m-n}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}\frac{{\left(3125\right)}^2}{15625}=\frac{5^{10}}{5^6}\\ =5^{10-6}\\ =5^4\\ =625\end{array}$

Thus, the simplified form of the expression $\frac{{\left(3125\right)}^2}{15625}$ is $625$ .