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:

  1. When you see the purple blocks (definitions, theorems) or the yellow blocks (facts, methods) in the lecture notes (or during lecture), what should you do with them to prepare for in-class assessments?
  2. Consider the mathematical entity $x - 2$.
    1. In the global context, do we have terms or factors?
    2. How can we modify the expression (but maintain equality) to convert the expression so it's simultaneously a term and a factor in the global context?
  3. For each of the following expressions, list all possible terms/factors and their contexts. Use the table!
    1. $3x - 2$
    2. $(2x - 1)(x + 2) + z$
    3. $(x - y)(2y - z) + \left[(x-4)(3x-3) - z\right]2 - 1$
      Hint: There are six context levels.
    4. $4y^2 - 5y^3(x-2)^2$
      Hint: you can treat exponentiated factors, for example $y^2$, as a singular factor (provided you are in the correct context).
    5. $3(x-1)^2(2x+3)^3 - 4(x-1)^3(2x+3)^2$
  4. In the expression \[-42(x^2 + 3)(x+3)^2 + 5(x+4)^4(x-3)^2\] under which local context level is $(x^2 + 3)$ considered a factor?
  5. In fraction property #5, which says \[\dfrac{ac}{bc} = \dfrac{a}{b}\] what must $c$ need to be in order to be cancelled out?
  6. Can I cross out the $x^2$ in \[\dfrac{x^2 + 1}{x^2 + 2}\] to get $\dfrac{1}{2}$? Give the reason why or why not.
  7. Can I cross out the $x - 1$ in \[\dfrac{(x-1)(x+2)}{(x-1)(x+3)}\] to get $\dfrac{x+2}{x+3}$? Give the reason why or why not.
  8. Can I cross out the $x - 1$ and $x + 3$ in \[\dfrac{(x-1)(x+2) + 4(x+3)}{(x-1)(x+3)}\] to get $\dfrac{x+2 + 4}{1}$? Give the reason why or why not.
  9. Can I cross out the $(x+2)$ in \[\dfrac{3(x+2)(x-1) - (x+3)(x-2)}{(x-1)(x+2)}\] to get $\dfrac{3(x-1) - (x+3)(x-2)}{(x-1)}$? Give the reason why or why not.
  10. Write down one fractional expression which satisfies the following. Multiple answers are possible.
    • Global context of numerator comprises of three terms
    • Global context of denominator comprieses of two terms
    • Each L1 term in the numerator contains two L2 factors
    • Each L1 term in the denominator contains three L2 factors
    • In the numerator, the first local (L2) factor in the first global term contains five terms
  11. Use fraction laws to simplify the following:
    1. $\dfrac{4}{7}\div 3$
    2. $\dfrac{3}{5} \div \dfrac{6}{5}$
    3. $2 \cdot \dfrac{1}{3} \cdot \dfrac{5}{7}$
    4. $\dfrac{3}{5} + \dfrac{4}{15}$
    5. $\dfrac{5}{12} + \dfrac{7}{30}$
    6. $\dfrac{11}{40} + \dfrac{5}{24}$
  12. Which statements are true? Point out which property was used (possibly incorrectly).
    1. $2 + 5 \cdot 3= 5 + 2 \cdot 3$
    2. $y(x-1) = (x-1)y$
    3. $x(x+y) = x\cdot x + x\cdot y$
    4. $(a + b)(c + d) = (a + b)c + (a + b)d$
    5. $2 - \left[(x-3)4\right] = 2 - (4x - 12)$
  13. 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?
  14. Graph the interval on the real line:
    1. $(-8, -2) \cup (-1, 1)$
    2. $(-\infty, 3) \cup (3, \infty)$
    3. $(-\infty, -1) \cup [2, 4] \cup (5, \infty)$
    4. $[1, 3) \cup (3, \infty)$
  15. Let $x \in \mathbb{R}$. In English, what does $\lvert x \rvert$ mean?
    Note:
    • $\mathbb{R}$ is the set of all real numbers.
    • The symbol $\in$ means "to be an element of".
    • So $x \in \mathbb{R}$ means "$x$ is an element of the real numbers."
  16. Simplify the following:
    1. $-(x - 3)$
    2. $-(1 - x) - (x - 2)$
    3. $\lvert -3 \rvert$
    4. $\lvert - \lvert 3\rvert \rvert$
    5. $\lvert -\lvert -3\rvert \rvert$
    6. $\lvert 1 - \pi \rvert$
  17. 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?
  18. 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?
  19. 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?
  20. When asked to "simplify" a problem involving exponents, when is the problem considered fully simplified?
    Solution: You need to
    1. Use all exponent laws until you cannot.
    2. Get rid of all negative exponents.
    3. Try to leave the expression in global factors.
    4. If you have fractions, the answer needs to be one fraction.
  21. Simplify the following:
    1. $-2^3$
    2. $(-2)^3$
    3. $\left(\dfrac{1}{2}\right)^{-1}$
    4. $36^{\frac{1}{2}}$
    5. $6^{\frac{3}{2}}$
    6. $4^{-\frac{3}{2}}$
    7. $81^{-\frac{1}{4}}$
    8. $\left(\dfrac{4}{9}\right)^{-1/2}$
  22. Use Exponent laws to simplify the following.
    I advise (not required) calling out each exponent law as you use them to help you memorize them.
    1. $x^{-2}$
    2. $x^4 \cdot x^7$
    3. $(3x^2y)^3$
    4. $\left[\dfrac{(x^3y^5)^4}{z^{-4}}\right]^0$
    5. $\dfrac{y}{(x-2)^{-1}}$
    6. $\left(\dfrac{x}{y}\right)^2 \cdot \left(\dfrac{y^2}{z}\right)^3$
    7. $\dfrac{(x+1)(x+2)}{(x+1)^{-2}(x+2)^2(x+3)}$
    8. $\dfrac{(x-1)(x+1)^{-3}(x-2)}{[2(x-1)]^{-1}(x+1)^2(x-2)^{-2}}$
    9. $\left(\dfrac{x(x-1)^2}{(x-1)^{-3}x^2}\right)^{-2}$

    23. The remaining problems will be on next week's homework. Skip for now.

  23. You tried converting $\sqrt[5]{x^3}$ into $x^{\frac{5}{3}}$. Is this wrong?
  24. Rewrite each root as an exponent and fully simplify.
    1. $\dfrac{\sqrt[3]{x^2}}{x^{2/3}}$
    2. $\dfrac{(-3)^4\sqrt{x}}{3^2\sqrt[3]{x}}$
    3. $\sqrt[4]{16x^8}$
    4. $\sqrt[4]{x^3}\cdot \sqrt{x}$
    5. $\dfrac{\sqrt[3]{(2x+1)^2}(x-2)^{2/5}}{\sqrt[4]{(2x+1)^5}}$