Question

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.