Some Algebra Practice questions          Back to Math 161

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  1. Evaluate each expression without using a calculator.  solution

    2. (a) \((-3)^{4}\)   (b) \(-3^{4}\)   (c) \(3^{-4}\)   (d) \(\frac{5^{23}}{5^{21}}\) (e) \(\left(\frac{2}{3}\right)^{-2}\)  (f) \(16^{-3 / 4}\)

  2. Simplify each expression. Write your answer without negative exponents. solution

    3. (a) \(\sqrt{200}-\sqrt{32}\)   (b) \(\left(3 a^{3} b^{3}\right)\left(4 a b^{2}\right)^{2}\)   (c) \(\left(\dfrac{3 x^{3 / 2} y^{3}}{x^{2} y^{-1 / 2}}\right)^{-2}\)

  3. Expand and simplify. solution

    4. (a) \(3(x+6)+4(2 x-5)\)   (b) \((x+3)(4 x-5)\)   (c) \((\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})\)   (d) \((2 x+3)^{2}\)  (e) \((x+2)^{3}\)

  4. Multiply or divide to simplify as a sum/difference of terms.  solution

    5. (a) \( \sqrt{x} (x^2-1) \)     (b) \(\dfrac{\sqrt{x}+x}{x^2} \)     (c) \(\dfrac{t^2-3t+\sqrt[3]t}{\sqrt{t}} \)

  5. Factor the expression.  solution

    6. (a) \(x^2+3x \)   (b) \(2x^3+4x \)   (c) \( x^3-4x \)   (d) \( x^3+9x \)   (e) \(x^2-5x+6 \)   (f) \(x^2-x-6 \)

  6. Factor the expression.   solution

    7. (a) \(x^2+7x+3 \)   (b) \(x^2-x-6 \)   (c) \(6x^2+x-1\)   (d) \(3x^2-14x+8 \)

     

    Solving Quadratic Equations (by factoring, completing square, formula)

     

  7. Solve the following equations.   solution

    8. (a) \(1-\dfrac{1}{x^2}=0 \)   (b) \(\sqrt{x-1}+3=5 \)   (c) \(\sqrt{x}+2=x \)   (d) \(|2x-3|=4 \)

  8. Simplify the rational expression. solution

    9. (a) \(\dfrac{x^{2}+3 x+2}{x^{2}-x-2}\)     (b) \(\dfrac{2 x^{2}-x-1}{x^{2}-9} \cdot \dfrac{x+3}{2 x+1}\)    (c) \(\dfrac{\frac{y}{x}-\frac{x}{y}}{\frac{1}{y}-\frac{1}{x}}\)

  9. Simplify each expression.  solution

    10. (a) \(\dfrac{x^3-9x}{2x-6}\)     (b) \(\dfrac{\frac{1}{2+h}-\frac{1}{2}}{h} \)     (c) \(\displaystyle{ \frac{(5+h)^2-25}{h}} \)

  10. Rationalize the numerator.  solution

    11. (a) \(\dfrac{\sqrt{9+h}-3}{h} \)    (b) \(\dfrac{\sqrt{x}-2}{x-4}\)

  11. Evaluate the difference quotient as indicated.

    12. (a) \( \dfrac{f(a+h)-f(a)}{h}\)   for   \( f(x)=x^2\)  solution

    (b) \( \dfrac{f(4+h)-f(4)}{h}\)   for   \( f(x)=\sqrt{x}\) solution

    (c) \( \dfrac{f(x+h)-f(x)}{h}\)   for   \( f(x)=\dfrac{1}{\sqrt{x}}\)  solution

  12. Decompse the rational function as a sum of partial fractions.   solution

    13. (a) \(\dfrac{6}{x^2-9} \)     (b) \(\dfrac{x^2-2x}{x-1} \)    (c) \(\dfrac{x}{x^2+x-2}\)

  13. Compute the ratio \(\left| \dfrac{a_{n+1}}{a_n}\right| \), and simplify.   solution

    14. (a) \(a_n = (-1)^n \dfrac{1}{2n+3} \)     (b) \(a_n = \dfrac{(-2)^n}{n!} \)    (c) \(a_n = (-1)^{n-1} \dfrac{1}{(2n+1)!} \)