Sure! Let's solve the problem step-by-step.
Given the expression:
[tex]\[
(8^4)^{-2}
\][/tex]
We need to simplify this to a single power of 8. To do this, we can apply the power of a power rule from the laws of exponents. The power of a power rule states:
[tex]\[
(a^m)^n = a^{m \cdot n}
\][/tex]
Here, our base [tex]\(a\)[/tex] is 8, [tex]\(m\)[/tex] is 4, and [tex]\(n\)[/tex] is -2. Applying the power of a power rule:
[tex]\[
(8^4)^{-2} = 8^{4 \cdot -2}
\][/tex]
Now, multiply the exponents:
[tex]\[
4 \times -2 = -8
\][/tex]
Thus, we can simplify the expression to:
[tex]\[
(8^4)^{-2} = 8^{-8}
\][/tex]
The combined exponent is -8.
Next, we need to calculate the numerical value of [tex]\(8^{-8}\)[/tex]. When an exponent is negative, it signifies the reciprocal of the base raised to the positive exponent:
[tex]\[
8^{-8} = \frac{1}{8^8}
\][/tex]
We already know the result of this calculation is:
[tex]\[
8^{-8} = 5.960464477539063 \times 10^{-8}
\][/tex]
or equivalently,
[tex]\[
8^{-8} = 5.960464477539063e-08
\][/tex]
So, the final result is that:
[tex]\[
\left(8^4\right)^{-2} \text{ can be written as a single power of 8: } 8^{-8}
\][/tex]
And the value of [tex]\(8^{-8}\)[/tex] is approximately [tex]\(5.960464477539063 \times 10^{-8}\)[/tex].
