To simplify the expression [tex]\(\left(-5 g^2 h^6\right)^3\)[/tex], we need to apply the rules of exponents:
1. Simplify the coefficient: We start with [tex]\(-5\)[/tex]. Raising [tex]\(-5\)[/tex] to the power of 3, we get [tex]\((-5)^3\)[/tex].
2. Simplify the exponent for [tex]\(g\)[/tex]: The original exponent on [tex]\(g\)[/tex] is 2. When raised to a power of 3, we multiply the exponents: [tex]\(2 \times 3\)[/tex].
3. Simplify the exponent for [tex]\(h\)[/tex]: The original exponent on [tex]\(h\)[/tex] is 6. When raised to a power of 3, we also multiply these exponents: [tex]\(6 \times 3\)[/tex].
Step-by-step:
1. Coefficient:
[tex]\[ (-5)^3 = -125 \][/tex]
2. Exponent for [tex]\(g\)[/tex]:
[tex]\[ (g^2)^3 = g^{2 \times 3} = g^6 \][/tex]
3. Exponent for [tex]\(h\)[/tex]:
[tex]\[ (h^6)^3 = h^{6 \times 3} = h^{18} \][/tex]
Putting it all together, the expression simplifies to:
[tex]\[ \left(-5 g^2 h^6\right)^3 = -125 g^6 h^{18} \][/tex]
Thus, the simplified expression is:
[tex]\[ -125 g^6 h^{18} \][/tex]
So, the correct answer is [tex]\(\boxed{-125 g^6 h^{18}}\)[/tex].
