**Directions:**

- Show each step of your work and fully simplify each expression.
- Turn in your answers in class on a physical piece of paper.
**Staple multiple sheets together.**- Feel free to use Desmos for graphing.

Answer the following:

Simplify problems 1-15.

- $x^4 \cdot x^7$
- $\left(\dfrac{1}{2}\right)^{-1}$
- $(x^2y)^3$
- $\left[\dfrac{(x^3y^5)^4}{z^{-4}}\right]^0$
- $\left(\dfrac{x}{y}\right)^2 \cdot \left(\dfrac{y^2}{z}\right)^3$
- $\dfrac{xy^{-3}z}{(2x)^{-1}y^2z^{-2}}$
- $\dfrac{(x+1)(x+2)}{(x+1)^{-2}(x+2)^2(x+3)}$
- $\dfrac{1}{3} + \dfrac{1}{6}$
- $\dfrac{(x + 2)}{2(x+2)}$
- $\dfrac{(x+1)(x + 2)}{2(x+3)(x+2)^2}$
- $\dfrac{1}{x+1} + \dfrac{1}{(x+1)^2}$
- $\dfrac{1}{(x+1)(x+2)} + \dfrac{1}{(x+1)^2}$
- $\dfrac{1}{x} + \dfrac{1}{x^2} + \dfrac{1}{x^3}$
- $x + 2x + 3x$
- $4xy^2 + 5x^2y - 3xy^2 - 4x^2y$

- Your friend tells you "Anytime you write an equals sign to perform a mathematical move, you should be able to call out which law or property you used." Is this true or false?
- If \[A = \{1,2,3,4,5,6,7\} \qquad B = \{2,4,6,8\} \qquad C = \{7,8,9,10\}\] find the following:
- $A\cup B$
- $B\cup C$
- $A \cap C$

- Simplify and graph the resulting set:
- $(-2, 0) \cup (-1, 1)$
- $(-2, 0] \cap (-1, 1)$
- $(-\infty, 6]\cap (2, 10)$
- $(-\infty, -4]\cup (4, \infty)$

- A student tries to simplify $x^2 + x^3$ by applying exponent laws. They write \[x^2 + x^3 = x^{2+3} = x^5\] Why is this incorrect?
- A student tries to simplify $x^2 \cdot x^3$ by applying exponent laws. They write \[x^2 \cdot x^3 = x^{2\cdot3} = x^6\] Why is this incorrect?
- A student tries to simplify $(a + b)^2$ by applying exponent laws. They write \[(a + b)^2 = a^2 + b^2\] Why is this incorrect?
- What is the difference between a
**term**and a**factor**? - A student tries to simplify $\frac{3x}{x^2 + 3x + 1}$ by cancelling common terms. Noticing the $3x$ is shared, they write \[\frac{3x}{x^2 + 3x + 1} =\frac{1}{x^2 + 1}\] Why is this incorrect?
- The distributive law says \[a(b+c) = ab + ac\]Your friend tries to simplify $(x + 3)(x + 2)$ by using the Distributive law: \[(x + 3)(x + 2) = x + 3 \cdot x + x + 3 \cdot 2\] What did they do wrong?
- Suppose I have an expression $x(y^2 + y)$ and I am asked to expand it. I try $x\cdot y^2 + x \cdot y = xy^2 + xy$. What property did I use?
- What is the relationship between factoring and expanding?
- Factor the following:
- $x^3 + x^2 + x$
- $3xy + 4xy^2 + 3xy$
- $(x+1)(x+2) + (x+2)(x+3)$
- $3(x+1)^2(x+2) - (x+1)(x+2)(x+3) + x(x+1)$

- In the expression $\dfrac{x}{x + 1}$, can we cancel out the $x$'s to get $\frac{0}{0 + 1} = 0$? If we cannot, explain why.
- For the expression $\dfrac{x + 1}{x}$, can we cancel out the $x$'s to get $\frac{0 + 1}{0}$ which is undefined? If we cannot, explain why.
- In the expression $\dfrac{x + 1}{x}$, can we rewrite this as $\dfrac{x}{x} + \dfrac{1}{x}$, which simplifies as $1 + \dfrac{1}{x}$? If we cannot, explain why.