Question

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

Answer 1

To rewrite the expression \( \frac{3^{17}}{8^{14}} \) with positive exponents only, we first recognize that we can apply the property of negative integer exponents, which states:

\[
a^{-n} = \frac{1}{a^n}
\]

Here, we can rewrite \( 8^{14} \) in the denominator using negative exponents:

\[
\frac{3^{17}}{8^{14}} = 3^{17} \cdot 8^{-14}
\]

Now, we also want to express \( 8 \) as a power of \( 2 \):

\[
8 = 2^3
\]

Thus, we can rewrite \( 8^{14} \) as:

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

Now substituting this into our earlier expression yields:

\[
3^{17} \cdot 8^{-14} = 3^{17} \cdot (2^{42})^{-1} = 3^{17} \cdot 2^{-42}
\]

At this point, the expression is:

\[
3^{17} \cdot 2^{-42}
\]

It’s helpful to write \( 2^{-42} \) in the numerator:

\[
3^{17} \cdot 2^{-42} = \frac{3^{17}}{2^{42}}
\]

Thus, we have:

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

And since we started with negative exponents, we now have the final expression:

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

This results in an expression equivalent to \( \frac{3^{17}}{8^{14}} \) with positive exponents only as:

\[
3^{17} \cdot 2^{-42}
\]

So, the final result can also be written as:

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

or:

\[
3^{17} \cdot 2^{-42}
\]

This completes the transformation!