Let's go through each problem from your worksheet and simplify them step-by-step.

1) $3xy + 4xy + 5x^2y + 6xy^2$

Combine like terms:

• $3xy + 4xy = 7xy$

The expression becomes: $7xy + 5x^2y + 6xy^2$ This is simplified as much as possible.

2) $6(x^2 + y^2) - 7(x^2 + y^2)$

Factor out the common term $(x^2 + y^2)$: $(6 - 7)(x^2 + y^2) = -1(x^2 + y^2)$ The simplified expression is: $-(x^2 + y^2)$

3) $\frac{18x^2y^2}{24xy^2}$

Simplify by canceling out common terms:

• $18x^2y^2 \div 24xy^2 = \frac{18}{24} \times \frac{x^2}{x} \times \frac{y^2}{y^2}$

Simplify further: $\frac{3}{4} \times x = \frac{3x}{4}$

4) $x^2(-2xz)(4z^2)$

Multiply the terms:

• First, multiply $(-2xz)$ and $4z^2$: $(-2 \times 4) \times (xz \times z^2) = -8xz^3$
• Then multiply by $x^2$: $x^2 \times (-8xz^3) = -8x^3z^3$ The simplified expression is: $-8x^3z^3$

5) $5t + 3r + 9t - 10r - 8$

Combine like terms:

• $5t + 9t = 14t$
• $3r - 10r = -7r$

The simplified expression is: $14t - 7r - 8$

6) $(-3ab)(-3ab^2)(-3a^2b^3)$

Multiply the terms: $(-3ab) \times (-3ab^2) \times (-3a^2b^3)$ Multiply coefficients: $(-3) \times (-3) \times (-3) = -27$ Multiply variables: $a \times a \times a^2 = a^4 \quad \text{and} \quad b \times b^2 \times b^3 = b^6$ The simplified expression is: $-27a^4b^6$

7) $\frac{-5x^7y^4z^3}{13xy^2z}$

Simplify by canceling out common terms: $\frac{-5x^7y^4z^3}{13xy^2z} = \frac{-5}{13} \times \frac{x^7}{x} \times \frac{y^4}{y^2} \times \frac{z^3}{z}$ Simplify further: $\frac{-5x^6y^2z^2}{13}$

8) $(3pq^2r^3)(\frac{1}{3q^4r})$

Simplify by multiplying: $3pq^2r^3 \times \frac{1}{3q^4r} = \frac{3}{3} \times p \times \frac{q^2}{q^4} \times \frac{r^3}{r}$ Simplify further: $p \times \frac{1}{q^2} \times r^2 = \frac{pr^2}{q^2}$

These are the simplified forms of each expression! Let me know if you'd like a detailed explanation of any step or have further questions.

Here are 5 follow-up questions to deepen your understanding:

1. How do you determine like terms when combining algebraic expressions?
2. What properties of exponents did we use in simplifying the expressions?
3. How can factoring help in simplifying algebraic expressions?
4. What strategies can be used to simplify fractions involving variables?
5. What is the importance of canceling terms in rational expressions?

Tip: Always look for common factors or terms that can be grouped or canceled out to simplify complex expressions efficiently.