Question

Solve the problem involving exponents 2/3⋅6−(6−4)3%0D%0A2%0D%0A3%0D%0A⋅%0D%0A6%0D%0A−%0D%0A(%0D%0A6%0D%0A−%0D%0A4%0D%0A)%0D%0A3%0D%0A.(1 point)

Answer 1

Let's solve the expression step by step. The expression is:

\[
\frac{2}{3} \cdot 6^{-(6-4)^3}
\]

First, simplify the exponent in the expression:

\[
6 - 4 = 2
\]

Now substituting this into the exponent:

\[
(6 - 4)^3 = 2^3 = 8
\]

Next, substitute this back into the expression:

\[
\frac{2}{3} \cdot 6^{-8}
\]

This can be rewritten as:

\[
\frac{2}{3} \cdot \frac{1}{6^8}
\]

Now, multiply the fractions together:

\[
\frac{2}{3 \cdot 6^8}
\]

6 can be factored as \(2 \cdot 3\), so:

\[
6^8 = (2 \cdot 3)^8 = 2^8 \cdot 3^8
\]

This means:

\[
\frac{2}{3 \cdot 6^8} = \frac{2}{3 \cdot (2^8 \cdot 3^8)} = \frac{2}{2^8 \cdot 3^9}
\]

Now simplifying this gives:

\[
\frac{1}{2^7 \cdot 3^8}
\]

Thus, the final answer for the expression is:

\[
\frac{1}{2^7 \cdot 3^8}
\]

or approximately:

\[
\frac{1}{128 \cdot 6561}
\]

If you calculate \(128 \cdot 6561\), you get 8388608, leading us to:

\[
\frac{1}{8388608}
\]

This is the simplified value of the given expression.