Homework 1

Directions:

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

Answer the following:

Simplify problems 1-15.

  1. $x^4 \cdot x^7$
  2. $\left(\dfrac{1}{2}\right)^{-1}$
  3. $(x^2y)^3$
  4. $\left[\dfrac{(x^3y^5)^4}{z^{-4}}\right]^0$
  5. $\left(\dfrac{x}{y}\right)^2 \cdot \left(\dfrac{y^2}{z}\right)^3$
  6. $\dfrac{xy^{-3}z}{(2x)^{-1}y^2z^{-2}}$
  7. $\dfrac{(x+1)(x+2)}{(x+1)^{-2}(x+2)^2(x+3)}$
  8. $\dfrac{1}{3} + \dfrac{1}{6}$
  9. $\dfrac{(x + 2)}{2(x+2)}$
  10. $\dfrac{(x+1)(x + 2)}{2(x+3)(x+2)^2}$
  11. $\dfrac{1}{x+1} + \dfrac{1}{(x+1)^2}$
  12. $\dfrac{1}{(x+1)(x+2)} + \dfrac{1}{(x+1)^2}$
  13. $\dfrac{1}{x} + \dfrac{1}{x^2} + \dfrac{1}{x^3}$
  14. $x + 2x + 3x$
  15. $4xy^2 + 5x^2y - 3xy^2 - 4x^2y$
  1. 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?
  2. If \[A = \{1,2,3,4,5,6,7\} \qquad B = \{2,4,6,8\} \qquad C = \{7,8,9,10\}\] find the following:
    1. $A\cup B$
    2. $B\cup C$
    3. $A \cap C$
  3. Simplify and graph the resulting set:
    1. $(-2, 0) \cup (-1, 1)$
    2. $(-2, 0] \cap (-1, 1)$
    3. $(-\infty, 6]\cap (2, 10)$
    4. $(-\infty, -4]\cup (4, \infty)$
  4. 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?
  5. 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?
  6. 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?
  7. What is the difference between a term and a factor?
  8. 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?
  9. 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?
  10. 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?
  11. What is the relationship between factoring and expanding?
  12. Factor the following:
    1. $x^3 + x^2 + x$
    2. $3xy + 4xy^2 + 3xy$
    3. $(x+1)(x+2) + (x+2)(x+3)$
    4. $3(x+1)^2(x+2) - (x+1)(x+2)(x+3) + x(x+1)$
  13. 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.
  14. 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.
  15. 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.