Question

use the power rule of exponents to simplify 9^3 ^6 in exponent form

Answer 1

To simplify \( 9^{3^6} \) using the power rule of exponents, we can use the property that states \( (a^m)^n = a^{m \cdot n} \).

In this case, we have:

\[
9^{3^6} = 9^{(3^6)} = 9^{3^{6}}
\]

To rewrite \( 9 \) as an exponent with base \( 3 \), we know that \( 9 = 3^2 \). Thus:

\[
9^{3^6} = (3^2)^{3^6}
\]

Now, applying the power rule:

\[
(3^2)^{3^6} = 3^{2 \cdot 3^6}
\]

Now, \( 2 \cdot 3^6 \) can be left as is, or further simplified. The expression \( 2 \cdot 3^6 \) doesn't have any particular simplification in terms of exponentiation since it is already in an exponent form.

Thus, the final simplified result is:

\[
3^{2 \cdot 3^6}
\]

If you'd like a numerical exponent calculation, you could evaluate \( 3^6 \) first:

\[
3^6 = 729
\]

So the exponent becomes \( 2 \cdot 729 = 1458 \). Therefore, in numerical form:

\[
9^{3^6} = 3^{1458}
\]

However, in exponent form, the simplified answer is:

\[
3^{2 \cdot 3^6}
\]