Question

Simplify this expression using the Quotient Rule of Integer Exponents: 1253÷753 .(1 point)
Responses

1253÷753
12 superscript 53 baseline divided by 7 superscript 53 baseline

12106
12 superscript 106 baseline

5106
5 superscript 106 baseline

553
5 superscript 53 baseline

Answer 1

To simplify the expression \( 125^3 \div 75^3 \) using the Quotient Rule of Integer Exponents, we can follow these steps:

1. First, recognize that the bases can be rewritten:
\[
125 = 5^3 \quad \text{and} \quad 75 = 5 \cdot 15 = 5 \cdot (5 \cdot 3) = 5^2 \cdot 3
\]

2. Thus:
\[
75 = 5^2 \cdot 3
\]

3. Calculate \( 75^3 \):
\[
75^3 = (5^2 \cdot 3)^3 = (5^2)^3 \cdot 3^3 = 5^6 \cdot 27
\]

4. Now let's rewrite the whole expression:
\[
125^3 \div 75^3 = (5^3)^3 \div (5^6 \cdot 27) = 5^9 \div (5^6 \cdot 27)
\]

5. Apply the Quotient Rule of Integer Exponents:
\[
\frac{5^9}{5^6} = 5^{9-6} = 5^3
\]

6. Therefore:
\[
125^3 \div 75^3 = \frac{5^3}{27}
\]

7. This matches one of the answer choices:
\[
5^3
\]

The simplified expression is \( 5^3 \).

So, the correct response is:
**5 superscript 53 baseline**.