Certainly! Let's simplify each of the expressions step-by-step using the power law.
### A. Simplify (9^4)^12
To simplify the expression [tex]\((9^4)^{12}\)[/tex]:
1. Apply the power of a power rule, which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(9^4)^{12} = 9^{4 \cdot 12}
\][/tex]
2. Calculate the new exponent:
[tex]\[
4 \cdot 12 = 48
\][/tex]
3. Simplify the expression:
[tex]\[
9^{48}
\][/tex]
### B. Simplify (4^{13})^{12}
To simplify the expression [tex]\((4^{13})^{12}\)[/tex]:
1. Apply the power of a power rule, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(4^{13})^{12} = 4^{13 \cdot 12}
\][/tex]
2. Calculate the new exponent:
[tex]\[
13 \cdot 12 = 156
\][/tex]
3. Simplify the expression:
[tex]\[
4^{156}
\][/tex]
### C. Simplify (9^4)^{-91}
To simplify the expression [tex]\((9^4)^{-91}\)[/tex]:
1. Apply the power of a power rule, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(9^4)^{-91} = 9^{4 \cdot (-91)}
\][/tex]
2. Calculate the new exponent:
[tex]\[
4 \cdot (-91) = -364
\][/tex]
3. Simplify the expression:
[tex]\[
9^{-364}
\][/tex]
### D. Simplify (5^1)^3
To simplify the expression [tex]\((5^1)^3\)[/tex]:
1. Apply the power of a power rule, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(5^1)^3 = 5^{1 \cdot 3}
\][/tex]
2. Calculate the new exponent:
[tex]\[
1 \cdot 3 = 3
\][/tex]
3. Simplify the expression:
[tex]\[
5^3 = 125
\][/tex]
So, the simplified expressions are:
1. [tex]\((9^4)^{12} = 9^{48}\)[/tex]
2. [tex]\((4^{13})^{12} = 4^{156}\)[/tex]
3. [tex]\((9^4)^{-91} = 9^{-364}\)[/tex]
4. [tex]\((5^1)^3 = 5^3 = 125\)[/tex]
Below are the final results:
1. [tex]\(9^{48}\)[/tex]
2. [tex]\(4^{156}\)[/tex]
3. [tex]\(9^{-364}\)[/tex]
4. [tex]\(125\)[/tex]
