Which values of $a$ and $b$ make the following equation true?
$\left(5 x^7 y^2\right)\left(-4 x^4 y^5\right)=-20 x^a y^b$
A. $a=11, b=7$
B. $a=11, b=10$
C. $a=28, b=7$
D. $a=28, b=10$
Answer (1)
Multiply the terms on the left side of the equation: ( 5 x 7 y 2 ) ( − 4 x 4 y 5 ) .
Apply the exponent rule x m ⋅ x n = x m + n to get − 20 x 7 + 4 y 2 + 5 .
Simplify the expression to − 20 x 11 y 7 .
Compare the exponents to find a = 11 and b = 7 , so the final answer is a = 11 , b = 7 .
Explanation
Understanding the Problem We are given the equation ( 5 x 7 y 2 ) ( − 4 x 4 y 5 ) = − 20 x a y b . Our goal is to find the values of a and b that make this equation true.
Multiplying the Terms First, let's multiply the terms on the left side of the equation. We have ( 5 x 7 y 2 ) ( − 4 x 4 y 5 ) .
Applying Exponent Rules Now, we multiply the coefficients and use the exponent rule x m x n = x m + n to simplify the expression. So, we have 5 ( − 4 ) x 7 + 4 y 2 + 5 .
Simplifying the Equation This simplifies to − 20 x 11 y 7 . Now we can rewrite the original equation as − 20 x 11 y 7 = − 20 x a y b .
Finding the Values of a and b By comparing the exponents of x and y on both sides of the equation, we can find the values of a and b . We see that a = 11 and b = 7 .
Final Answer Therefore, the values of a and b that make the equation true are a = 11 and b = 7 .
Examples
Understanding how to manipulate exponents is crucial in many fields, such as physics and computer graphics. For instance, when calculating the scaling of objects in a 3D game, you need to understand how the exponents of the scaling factors affect the final size of the object. If you scale an object by a factor of x in one dimension and y in another, the total scaling is proportional to x y , which is similar to the exponent rules we used in this problem. This ensures that the object is scaled correctly and appears as intended in the game.
