Question

use the laws of exponents to generate an equivalent expression. what is the new value of the base? what is the new value of the exponent? explain your thinking.

Explanation:

Step1: Recall exponent - division rule

When dividing two numbers with the same base \(a^m\div a^n=a^{m - n}\), and \(\sqrt{a}=a^{\frac{1}{2}}\). Here we have \(\frac{36^{\frac{5}{4}}}{36^{\frac{1}{4}}}\).

Step2: Apply the rule

Using the rule \(a^m\div a^n=a^{m - n}\), where \(a = 36\), \(m=\frac{5}{4}\), and \(n=\frac{1}{4}\), we get \(36^{\frac{5}{4}-\frac{1}{4}}\).

Step3: Calculate the exponent

\(\frac{5}{4}-\frac{1}{4}=\frac{5 - 1}{4}=\frac{4}{4}=1\). So the expression simplifies to \(36^1 = 36\), and we can rewrite \(36\) as \(6^2\).

Answer:

The new value of the exponent is \(1\), and the new value of the base is \(6\) (since \(36 = 6^2\)).