Which of the following is an equivalent expression to 14^-7/9^-13 with only positive exponents, generated by applying the Property of Negative Integer Exponents?

Question

1/9^13 times 14^-7
14^-7 times 9^13
14^7/9^13
9^13/14^7

Answer 1

To simplify the expression \( \frac{14^{-7}}{9^{-13}} \) using the Property of Negative Integer Exponents, we can rewrite the negative exponents as follows:

1. Rewrite \( 14^{-7} \) as \( \frac{1}{14^7} \).
2. Rewrite \( 9^{-13} \) as \( \frac{1}{9^{13}} \).

Applying these conversions to the original expression:

\[ \frac{14^{-7}}{9^{-13}} = \frac{\frac{1}{14^7}}{\frac{1}{9^{13}}} \]

When you divide by a fraction, it is equivalent to multiplying by its reciprocal, so we can rewrite it as:

\[ = \frac{1}{14^7} \times \frac{9^{13}}{1} = \frac{9^{13}}{14^7} \]

Thus, the equivalent expression with only positive exponents is:

\[ \frac{9^{13}}{14^7} \]

So, the correct answer from the options provided is:

9^13/14^7.

Answer 2

Which of the following is an equivalent expression to 7^3/25^-4 with only positive exponents, generated by applying the Property of Negative Integer Exponents?
7^3/25^4
25^4/7^-3
7^3 times 25^4
25^4/7^3

Answer 3

To simplify the expression \( \frac{7^3}{25^{-4}} \) and generate it with only positive exponents, we will apply the Property of Negative Integer Exponents.

According to this property, \( a^{-n} = \frac{1}{a^n} \). Therefore, we can rewrite \( 25^{-4} \) as \( \frac{1}{25^4} \).

Now, substituting this back into the expression:

\[ \frac{7^3}{25^{-4}} = 7^3 \div \frac{1}{25^4} = 7^3 \times 25^4 \]

This simplifies to:

\[ 7^3 \times 25^4 \]

Thus, the equivalent expression with only positive exponents is:

7^3 times 25^4.