To find the value of the expression [tex]\( 3^{-4} \)[/tex], we need to understand what the negative exponent means.
A negative exponent indicates that we take the reciprocal of the base raised to the positive exponent. Specifically, for any non-zero number [tex]\( a \)[/tex] and positive integer [tex]\( n \)[/tex],
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
For our specific problem, where the base is 3 and the exponent is -4, we apply this rule as follows:
[tex]\[
3^{-4} = \frac{1}{3^4}
\][/tex]
Next, we need to calculate [tex]\( 3^4 \)[/tex]. We find [tex]\( 3^4 \)[/tex] by multiplying 3 by itself four times:
[tex]\[
3^4 = 3 \times 3 \times 3 \times 3 = 81
\][/tex]
So,
[tex]\[
3^{-4} = \frac{1}{81}
\][/tex]
The value of [tex]\( \frac{1}{81} \)[/tex] when expressed as a decimal is approximately:
[tex]\[
0.012345679012345678
\][/tex]
Therefore, the value of the expression [tex]\( 3^{-4} \)[/tex] is:
[tex]\[
0.012345679012345678
\][/tex]
