Let's simplify the given expressions step-by-step.
### Expression 1: [tex]\(\left(g^4 h\right)^6\)[/tex]
First, we need to apply the power of a product rule, which states [tex]\((ab)^n = a^n b^n\)[/tex]:
[tex]\[
\left(g^4 h\right)^6 = \left(g^4\right)^6 \cdot \left(h\right)^6
\][/tex]
Next, apply the power of a power rule, which states [tex]\(\left(a^m\right)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
\left(g^4\right)^6 = g^{4 \cdot 6} = g^{24}
\][/tex]
[tex]\[
\left(h\right)^6 = h^6
\][/tex]
Therefore, the simplified form of [tex]\(\left(g^4 h\right)^6\)[/tex] is:
[tex]\[
g^{24} h^6
\][/tex]
### Expression 2: [tex]\(\left(g^{10} h\right)^6\)[/tex]
Similarly, apply the power of a product rule first:
[tex]\[
\left(g^{10} h\right)^6 = \left(g^{10}\right)^6 \cdot \left(h\right)^6
\][/tex]
Next, apply the power of a power rule:
[tex]\[
\left(g^{10}\right)^6 = g^{10 \cdot 6} = g^{60}
\][/tex]
[tex]\[
\left(h\right)^6 = h^6
\][/tex]
Therefore, the simplified form of [tex]\(\left(g^{10} h\right)^6\)[/tex] is:
[tex]\[
g^{60} h^6
\][/tex]
### Given Options:
1. [tex]\(g^{24} h\)[/tex]
2. [tex]\(g^{24} h^6\)[/tex]
3. [tex]\(g^{10} h\)[/tex]
4. [tex]\(g^{10} h^6\)[/tex]
### Compare the Simplified Expressions with the Options:
1. The simplified form of [tex]\(\left(g^4 h\right)^6\)[/tex] is [tex]\(g^{24} h^6\)[/tex], which corresponds to option 2.
2. The simplified form of [tex]\(\left(g^{10} h\right)^6\)[/tex] is [tex]\(g^{60} h^6\)[/tex], which does not correspond to any of the given options directly.
Thus, the correct answer that matches the simplified forms is:
[tex]\[
g^{24} h^6
\][/tex]
So, option 2 ([tex]\(g^{24} h^6\)[/tex]) is the simplified form of [tex]\(\left(g^4 h\right)^6\)[/tex].
