Question

Find the numerical equivalent of 3 over 9 ⋅ 3^−5

Answer 1

To find the numerical equivalent of \( \frac{3}{9} \cdot 3^{-5} \), we can simplify the expression step by step.

First, simplify \( \frac{3}{9} \):

\[
\frac{3}{9} = \frac{1}{3}
\]

Now, substituting this back into the original expression:

\[
\frac{1}{3} \cdot 3^{-5}
\]

Next, we can use the property of exponents which states that \( a^{-n} = \frac{1}{a^n} \):

\[
3^{-5} = \frac{1}{3^5}
\]

Thus, our expression becomes:

\[
\frac{1}{3} \cdot \frac{1}{3^5} = \frac{1}{3 \cdot 3^5}
\]

Using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we simplify the denominator:

\[
3 \cdot 3^5 = 3^{1+5} = 3^6
\]

So we have:

\[
\frac{1}{3^6}
\]

To find the numerical value of \( 3^6 \):

\[
3^6 = 729
\]

Therefore,

\[
\frac{1}{3^6} = \frac{1}{729}
\]

The numerical equivalent of the original expression \( \frac{3}{9} \cdot 3^{-5} \) is

\[
\boxed{\frac{1}{729}}
\]