Some Algebra Practice questions

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

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) \(3x^2-14x+8 \)

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.  solution

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

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