Some Algebra Practice questions          Back to Math 161

Properties of Exponents: Click here

Evaluate each expression without using a calculator.  solution

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

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

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

Expand and simplify. solution

(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) $(x+2)^{3}$

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

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

Factor the expression.  solution

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

Factor the expression.   solution

(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)

 

Solve the following equations.   solution

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

Simplify the rational expression. solution

(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) $\dfrac{\frac{y}{x}-\frac{x}{y}}{\frac{1}{y}-\frac{1}{x}}$

Simplify each expression.  solution

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

Rationalize the numerator.  solution

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

Evaluate the difference quotient as indicated.

(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

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

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

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

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