To determine the correct values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that make the equation true, we need to expand and simplify both sides of the equation:
Given equation:
[tex]\[
\left(5 x^7 y^2\right)\left(-4 x^4 y^5\right) = -20 x^a y^6
\][/tex]
We start by simplifying the left side of the equation. We will multiply the coefficients and use the property of exponents that [tex]\((x^m \cdot x^n = x^{m+n})\)[/tex] to combine the exponents.
1. Multiply the coefficients:
[tex]\[
5 \times (-4) = -20
\][/tex]
2. Combine the exponents of [tex]\( x \)[/tex]:
[tex]\[
x^7 \times x^4 = x^{7+4} = x^{11}
\][/tex]
3. Combine the exponents of [tex]\( y \)[/tex]:
[tex]\[
y^2 \times y^5 = y^{2+5} = y^{7}
\][/tex]
So, the left side of the equation becomes:
[tex]\[
-20 x^{11} y^7
\][/tex]
Next, we compare this to the right side of the equation:
[tex]\[
-20 x^a y^6
\][/tex]
For these two expressions to be equal, their coefficients, as well as the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], must be identical. Here, since the coefficients on both sides are already [tex]\(-20\)[/tex], let's match the exponents.
- Matching the exponents of [tex]\( x \)[/tex]:
[tex]\[
x^{11} = x^a \quad \Rightarrow \quad a = 11
\][/tex]
- Matching the exponents of [tex]\( y \)[/tex]:
[tex]\[
y^7 = y^6 \quad \Rightarrow \quad \text{This indicates a mistake, we correct the exponent here: } y^7 = y^b \quad \Rightarrow \quad b = 7
\][/tex]
Thus, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that make the given equation true are:
[tex]\[
\boxed{a = 11 \text{ and } b = 7}
\][/tex]
So the correct answer is [tex]\( (11, 7) \)[/tex] in the given choices.
