Question

1. $p^{2}q^{3}(p^{3}q^{2}-pq)$
2. $3xy(x^{2}+xy+y^{2})$
3. $5r^{2}s^{2}(-2r^{2}+3rs - 4s^{2})$
4. $3ab(4a^{2}-7b^{3})$
5. $-7c^{3}d^{6}(9c^{2}d^{8}-4cd^{6})$
6. $10m^{4}n(-2mn^{3}+3n)$

directions: distribute, then simplify the remaining expression. final answers must be in standard form.

1. $-2x(3x - 4)+7x$
2. $5w(-7w + 3)-2w(13 - 9w^{2})$

Explanation:

Let's solve each problem one by one using the distributive property (also known as the distributive law) and then simplify.

Problem 9: \( p^2q^3(p^3q^2 - pq) \)

Step 1: Distribute \( p^2q^3 \)

Using the distributive property \( a(b - c) = ab - ac \), we have:
\( p^2q^3 \cdot p^3q^2 - p^2q^3 \cdot pq \)

Step 2: Simplify using exponent rules (\( a^m \cdot a^n = a^{m + n} \))

For the first term: \( p^{2 + 3}q^{3 + 2} = p^5q^5 \)
For the second term: \( p^{2 + 1}q^{3 + 1} = p^3q^4 \)
So the simplified expression is \( p^5q^5 - p^3q^4 \)

Step 1: Distribute \( 3xy \)

Using the distributive property \( a(b + c + d) = ab + ac + ad \), we get:
\( 3xy \cdot x^2 + 3xy \cdot xy + 3xy \cdot y^2 \)

Step 2: Simplify using exponent rules

First term: \( 3x^{1 + 2}y = 3x^3y \)
Second term: \( 3x^{1 + 1}y^{1 + 1} = 3x^2y^2 \)
Third term: \( 3xy^{1 + 2} = 3xy^3 \)
So the simplified expression is \( 3x^3y + 3x^2y^2 + 3xy^3 \)

Step 1: Distribute \( 5r^2s^2 \)

Using the distributive property:
\( 5r^2s^2 \cdot (-2r^2) + 5r^2s^2 \cdot 3rs + 5r^2s^2 \cdot (-4s^2) \)

Step 2: Simplify using exponent rules

First term: \( -10r^{2 + 2}s^2 = -10r^4s^2 \)
Second term: \( 15r^{2 + 1}s^{2 + 1} = 15r^3s^3 \)
Third term: \( -20r^2s^{2 + 2} = -20r^2s^4 \)
So the simplified expression is \( -10r^4s^2 + 15r^3s^3 - 20r^2s^4 \)

Answer:

\( p^5q^5 - p^3q^4 \)

Problem 10: \( 3xy(x^2 + xy + y^2) \)