To solve the expression [tex]\(\frac{5}{4^{-2}}\)[/tex], we need to carefully handle the negative exponent and the fraction. Here's the detailed step-by-step solution:
1. Understand the Negative Exponent: A negative exponent means taking the reciprocal of the base and then raising it to the positive of that exponent.
For [tex]\(4^{-2}\)[/tex], we deal with the negative exponent first:
[tex]\[
4^{-2} = \frac{1}{4^2}
\][/tex]
2. Calculate the Positive Exponent:
Next, calculate [tex]\(4^2\)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
Now substitute this back into the expression for the negative exponent:
[tex]\[
4^{-2} = \frac{1}{16}
\][/tex]
3. Rewrite the Fraction:
Replace [tex]\(4^{-2}\)[/tex] with its equivalent fraction:
[tex]\[
\frac{5}{4^{-2}} = \frac{5}{\frac{1}{16}}
\][/tex]
4. Divide by a Fraction:
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of [tex]\(\frac{1}{16}\)[/tex] is [tex]\(16\)[/tex]:
[tex]\[
\frac{5}{\frac{1}{16}} = 5 \times 16
\][/tex]
5. Perform the Multiplication:
Finally, multiply the numerator by the reciprocal of the denominator:
[tex]\[
5 \times 16 = 80
\][/tex]
So, the expression [tex]\(\frac{5}{4^{-2}}\)[/tex] evaluates to 80.
As a summary:
- The numerator [tex]\(5\)[/tex] remains [tex]\(5\)[/tex].
- The denominator [tex]\(4^{-2}\)[/tex] translates to [tex]\(0.0625\)[/tex].
- The fraction value [tex]\(\frac{5}{0.0625}\)[/tex] results in [tex]\(80.0\)[/tex].
