Question

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

Answer 1

To rewrite the expression \(\frac{3^{17}}{8^{14}}\) with positive exponents only, we need to apply the property of negative integer exponents, which states that \(a^{-n} = \frac{1}{a^n}\).

First, we can express the denominator \(8^{14}\) with a negative exponent:

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

Now our expression is:

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

To further convert this to only positive exponents, we keep \(3^{17}\) as it is and can express \(8^{-14}\) in terms of positive exponent by writing:

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

However, to express it in a single term with positive exponent, we write it as:

\[
\frac{3^{17}}{8^{14}}
\]

Now, we need to ensure all bases have positive exponents only.

Since \(8 = 2^3\), we can rewrite \(8^{14}\) as:

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

Thus, our expression now becomes:

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

To write this using positive exponents, we can maintain this formatting. The final expression equivalent to \(\frac{3^{17}}{8^{14}}\) with positive exponents only is:

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

Alternatively, if you want to write it in a single fraction form with both terms in the numerator, it can be expressed as:

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

In summary, the expression \(\frac{3^{17}}{8^{14}}\) can be expressed with positive exponents as:

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

Or alternatively, keep it in the positive form as:

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