Using the properties of exponents, which of the following is an equivalent expression for √25/25^1/2? Could you explain why the answer is 1?

Answer 1

To simplify the expression \( \frac{\sqrt{25}}{25^{1/2}} \), let's first rewrite each component:

1. Expressing \( \sqrt{25} \): \[ \sqrt{25} = 25^{1/2} \]

2. Substituting back into the expression: \[ \frac{\sqrt{25}}{25^{1/2}} = \frac{25^{1/2}}{25^{1/2}} \]

3. Using the properties of exponents: The rule for dividing exponents with the same base states that \( \frac{a^m}{a^n} = a^{m-n} \). In this case: \[ \frac{25^{1/2}}{25^{1/2}} = 25^{1/2 - 1/2} = 25^0 \]

4. Evaluating \( 25^0 \): Any non-zero number raised to the power of zero is equal to 1: \[ 25^0 = 1 \]

Thus, the expression \( \frac{\sqrt{25}}{25^{1/2}} \) simplifies to 1, confirming that the equivalent expression is indeed 1.

Overall, the answer is 1 because (by the properties of exponents) you are dividing the same quantities (both are \( 25^{1/2} \)), which results in 1.