To solve this problem, let's go through each step carefully to determine the correct values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the given equation:
[tex]\[
(5x^7y^2)(-4x^4y^5) = -20x^a y^b
\][/tex]
1. Identify the coefficients:
- From the left-hand side of the equation:
[tex]\[
(5x^7y^2)(-4x^4y^5)
\][/tex]
The coefficients are 5 and -4.
2. Multiply the coefficients:
- Multiply these coefficients together:
[tex]\[
5 \times -4 = -20
\][/tex]
3. Combine the exponents of [tex]\(x\)[/tex]:
- We have [tex]\(x^7\)[/tex] and [tex]\(x^4\)[/tex]. When multiplying these terms, you add their exponents:
[tex]\[
x^7 \times x^4 = x^{7+4} = x^{11}
\][/tex]
4. Combine the exponents of [tex]\(y\)[/tex]:
- We have [tex]\(y^2\)[/tex] and [tex]\(y^5\)[/tex]. When multiplying these terms, you add their exponents:
[tex]\[
y^2 \times y^5 = y^{2+5} = y^7
\][/tex]
Putting it all together, we have:
[tex]\[
(5x^7y^2)(-4x^4y^5) = -20x^{11}y^7
\][/tex]
Comparing this with the right-hand side of the equation:
[tex]\[
-20x^a y^b
\][/tex]
We can see that:
[tex]\[
a = 11 \quad \text{and} \quad b = 7
\][/tex]
Therefore, the correct values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that make the equation true are:
[tex]\[
\boxed{a = 11, \, b = 7}
\][/tex]
