To solve the expression [tex]\((0.125)^{-3}\)[/tex], we need to evaluate the base [tex]\(0.125\)[/tex] raised to the power of [tex]\(-3\)[/tex]. Here’s a detailed step-by-step process:
1. Understanding Negative Exponents:
A negative exponent indicates that we take the reciprocal of the base and then raise it to the corresponding positive exponent. Therefore, [tex]\((0.125)^{-3}\)[/tex] is equivalent to:
[tex]\[
\left(\frac{1}{0.125}\right)^3
\][/tex]
2. Calculating the Reciprocal of the Base:
Next, we need to find the reciprocal of [tex]\(0.125\)[/tex]. The reciprocal of a number [tex]\(x\)[/tex] is [tex]\(\frac{1}{x}\)[/tex]. Thus, the reciprocal of [tex]\(0.125\)[/tex] is [tex]\(\frac{1}{0.125}\)[/tex].
Knowing the reciprocal:
[tex]\[
\frac{1}{0.125} = 8
\][/tex]
3. Raising to the Positive Exponent:
Now, we need to raise the reciprocal value to the power of 3:
[tex]\[
8^3
\][/tex]
4. Finding the Final Value:
To determine [tex]\(8^3\)[/tex], we multiply 8 by itself three times:
[tex]\[
8 \times 8 \times 8 = 512
\][/tex]
Therefore, the value of [tex]\((0.125)^{-3}\)[/tex] is:
[tex]\[
512
\][/tex]
