For all real numbers x and y, the expression (4x^3y^2)^2(2x^2y) is equivalent to which of … — SAT

For all real numbers $x$ and $y$ , the expression $(4x^3y^2)^2(2x^2y)$ is equivalent to which of the following?

The correct equivalent expression is

32x^8y^5

To find the equivalent expression, we first apply the power of a product rule to the first term:

(4x^3y^2)^2 = 4^2(x^3)^2(y^2)^2 = 16x^6y^4

. Next, we multiply this result by the second term:

(16x^6y^4)(2x^2y)

. Multiplying the coefficients gives

16 \times 2 = 32

. Adding the exponents of the same base variables gives

x^{6+2} = x^8

and

y^{4+1} = y^5

. This results in

32x^8y^5

Apply the power of a product rule to square the first expression:

(4x^3y^2)^2

16x^6y^4

When raising a product to a power, raise each factor to that power:

4^2 = 16

(x^3)^2 = x^{3 \times 2} = x^6

, and

(y^2)^2 = y^{2 \times 2} = y^4

Multiply the simplified first expression by the second expression:

(16x^6y^4)(2x^2y)

32x^8y^5

Multiply the numerical coefficients (

16 \times 2 = 32

) and add the exponents of variables with matching bases (

x^{6+2} = x^8

and

y^{4+1} = y^5

Simplifying equivalent algebraic expressions using rules of exponents

Estimated Time:45s

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