To rewrite the expression [tex]\( (-7)^{-2} \)[/tex] without using an exponent, we need to understand what a negative exponent signifies.
A negative exponent means that we take the reciprocal of the base raised to the corresponding positive exponent. Specifically, [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex].
So, applying this rule:
1. Start with the expression [tex]\( (-7)^{-2} \)[/tex].
2. Since the exponent is negative, we rewrite this as the reciprocal of [tex]\( (-7) \)[/tex] raised to the positive exponent 2:
[tex]\[
(-7)^{-2} = \frac{1}{(-7)^2}
\][/tex]
3. Now, calculate [tex]\( (-7)^2 \)[/tex]:
[tex]\[
(-7)^2 = (-7) \times (-7) = 49
\][/tex]
4. Substitute [tex]\( 49 \)[/tex] back into the reciprocal:
[tex]\[
\frac{1}{(-7)^2} = \frac{1}{49}
\][/tex]
Thus, the expression [tex]\( (-7)^{-2} \)[/tex] rewritten without an exponent is [tex]\(\frac{1}{49}\)[/tex].
The calculated numerical result of this fraction is approximately:
[tex]\[
\frac{1}{49} \approx 0.02040816326530612
\][/tex]
Therefore, [tex]\( (-7)^{-2} \)[/tex] simplifies to [tex]\(\frac{1}{49}\)[/tex] or approximately [tex]\( 0.02040816326530612 \)[/tex].
