Question

Question

Which of the following is an equivalent expression to 5^9 ⋅ 5^−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?

options:

5^9 over 5^−13

1 over 5^22

5^4

1 over 5^4

Answer 1

To simplify the expression \(5^9 \cdot 5^{-13}\) using the properties of integer exponents, we can use the property that states \(a^m \cdot a^n = a^{m+n}\).

Applying this property:

\[ 5^9 \cdot 5^{-13} = 5^{9 + (-13)} = 5^{-4} \]

Now we have \(5^{-4}\). To express this with only positive exponents, we can use the property \(a^{-m} = \frac{1}{a^m}\):

\[ 5^{-4} = \frac{1}{5^4} \]

Thus, an equivalent expression to \(5^9 \cdot 5^{-13}\) with only positive exponents is:

\[ \frac{1}{5^4} \]

The correct answer from the options provided is:

1 over 5^4