To simplify the exponential expression
[tex]\[
\frac{-12 x^7 y^7}{4 x^2 y^5}
\][/tex]
we follow these steps:
1. Simplify the coefficients:
The numerator coefficient is [tex]\(-12\)[/tex] and the denominator coefficient is [tex]\(4\)[/tex]. To simplify the coefficients, divide the numerator coefficient by the denominator coefficient:
[tex]\[
\frac{-12}{4} = -3
\][/tex]
2. Simplify the powers of [tex]\(x\)[/tex]:
The numerator has [tex]\(x\)[/tex] raised to the power of [tex]\(7\)[/tex] and the denominator has [tex]\(x\)[/tex] raised to the power of [tex]\(2\)[/tex]. To simplify [tex]\(x\)[/tex], subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
x^{7-2} = x^5
\][/tex]
3. Simplify the powers of [tex]\(y\)[/tex]:
The numerator has [tex]\(y\)[/tex] raised to the power of [tex]\(7\)[/tex] and the denominator has [tex]\(y\)[/tex] raised to the power of [tex]\(5\)[/tex]. To simplify [tex]\(y\)[/tex], subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
y^{7-5} = y^2
\][/tex]
Putting it all together, the simplified form of the given expression is:
[tex]\[
\frac{-12 x^7 y^7}{4 x^2 y^5} = -3 x^5 y^2
\][/tex]
Thus, the simplified exponential expression is:
[tex]\[
-3 x^5 y^2
\][/tex]
