Question

Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3^−17/8^−14
with positive exponents only.

Enter your answer as a fraction in simplest form.

Answer 1

To simplify the expression \( \frac{3^{-17}}{8^{-14}} \) using the property of negative integer exponents, we can rewrite the negative exponents as follows:

\[
3^{-17} = \frac{1}{3^{17}} \quad \text{and} \quad 8^{-14} = \frac{1}{8^{14}}
\]

Thus, our original expression becomes:

\[
\frac{3^{-17}}{8^{-14}} = \frac{1/3^{17}}{1/8^{14}} = \frac{1}{3^{17}} \times \frac{8^{14}}{1} = \frac{8^{14}}{3^{17}}
\]

Next, we simplify the expression. Since \(8\) can be written as \(2^3\), we have:

\[
8^{14} = (2^3)^{14} = 2^{42}
\]

Putting this back into our expression gives us:

\[
\frac{8^{14}}{3^{17}} = \frac{2^{42}}{3^{17}}
\]

Thus, the final expression with positive exponents only is:

\[
\frac{2^{42}}{3^{17}}
\]

To conclude, the equivalent expression is:

\[
\frac{2^{42}}{3^{17}}
\]