Chapter 7.4, Problem 24P

Reasoning The division property of exponents says that to simplify powers with the same base you subtract the exponents. Use examples to show why powers need to have the same base in order for this technique to work.

To determine

To explain:The division property of exponents says that to simplify the powers with the same base you subtract the exponents. Use examples to show why powers need to have same base in order for this technique to work.

Explanation of Solution

When bases are same then there is no ambiguity about which base we have to use while subtracting the exponents but when the bases of numerator and denominator are different then we don't know which base to use when we subtract the exponents.

For example: let $\frac{5^3}{6^1}$ different bases of numerator and denominator, then

  $\frac{5^3}{6^1}=\frac{125}{6}$

But if we apply the division property of exponents then we have 2 cases

case 1: $\frac{5^3}{6^1}=5^{\left(3-1\right)}=25\ne \frac{125}{6}$ , or

case 2: $\frac{5^3}{6^1}=6^{\left(3-1\right)}=36\ne \frac{125}{6}$

Hence, in both the cases we are not getting the required answer, so we cannot apply the rule to different bases.

Hence, we need same bases to apply the rule.