To find the expression equivalent to [tex]\(\left(12 g^2 h^7\right)\left(3 g^{-2} h^{-4}\right)\)[/tex], follow these steps:
1. Identify the coefficients and variables:
The given expression is [tex]\((12 g^2 h^7)(3 g^{-2} h^{-4})\)[/tex].
2. Multiply the coefficients separately:
The coefficients are [tex]\(12\)[/tex] and [tex]\(3\)[/tex]. Multiply these together:
[tex]\[
12 \times 3 = 36
\][/tex]
3. Apply exponent rules to variables with the same base:
- For the [tex]\(g\)[/tex] terms, combine the exponents using the rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[
g^2 \times g^{-2} = g^{2 + (-2)} = g^0
\][/tex]
Any number raised to the power of zero is 1, thus [tex]\(g^0 = 1\)[/tex].
- For the [tex]\(h\)[/tex] terms, combine the exponents the same way:
[tex]\[
h^7 \times h^{-4} = h^{7 + (-4)} = h^3
\][/tex]
4. Combine the results:
Multiplying the coefficients and the simplified variables together:
[tex]\[
36 \times 1 \times h^3 = 36 h^3
\][/tex]
Hence, the expression equivalent to [tex]\(\left(12 g^2 h^7\right)\left(3 g^{-2} h^{-4}\right)\)[/tex] is [tex]\(36 h^3\)[/tex].
Therefore, the correct answer is:
B. [tex]\(36 h^3\)[/tex]
