Question

5—8 multiply and simplify.

1. $8x^{2}y(3x^{3}y - 7xy + 8y)$

$24x^{5}y^{2}-56x^{3}y^{2}+64x^{2}y^{2}$
$-32x^{2}y + 64x^{2}y^{2}$

Explanation:

Step1: Distribute monomial to each term

$8x^2y \cdot 3x^3y - 8x^2y \cdot 7xy + 8x^2y \cdot 8y$

Step2: Calculate each product

For first term:

Multiply coefficients: $8 \times 3 = 24$
Multiply $x$ terms: $x^2 \times x^3 = x^{2+3} = x^5$
Multiply $y$ terms: $y \times y = y^{1+1} = y^2$
Result: $24x^5y^2$

For second term:

Multiply coefficients: $8 \times 7 = 56$
Multiply $x$ terms: $x^2 \times x = x^{2+1} = x^3$
Multiply $y$ terms: $y \times y = y^{1+1} = y^2$
Result: $56x^3y^2$ (keep the subtraction sign: $-56x^3y^2$)

For third term:

Multiply coefficients: $8 \times 8 = 64$
Multiply $x$ terms: $x^2$ (no additional $x$ term)
Multiply $y$ terms: $y \times y = y^{1+1} = y^2$
Result: $64x^2y^2$

Answer:

$24x^5y^2 - 56x^3y^2 + 64x^2y^2$