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Some Algebra Practice Questions

Evaluate or simplify the following expressions.

Properties of Exponents – Video Lesson

  1. Evaluate each expression without using a calculator. View Solution (a) \((-3)^{4}\)   (b) \(-3^{4}\)   (c) \(3^{-4}\)   (d) \(\dfrac{5^{23}}{5^{21}}\)   (e) \(\left(\dfrac{2}{3}\right)^{-2}\)   (f) \(16^{-3/4}\)
  2. Simplify each expression. Write your answer without negative exponents. View Solution (a) \(\sqrt{200}-\sqrt{32}\)   (b) \(\left(3a^{3}b^{3}\right)\left(4ab^{2}\right)^{2}\)   (c) \(\left(\dfrac{3x^{3/2}y^{3}}{x^{2}y^{-1/2}}\right)^{-2}\)
  3. Expand and simplify. View Solution (a) \(3(x+6)+4(2x-5)\)   (b) \((x+3)(4x-5)\)   (c) \((\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})\)   (d) \((2x+3)^{2}\)   (e) \((x+2)^{3}\)
  4. Multiply or divide to simplify as a sum or difference of terms. View Solution (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. View Solution (a) \(x^2+3x\)   (b) \(2x^3+4x\)   (c) \(x^3-4x\)   (d) \(x^3+9x\)   (e) \(x^2-5x+6\)   (f) \(x^2-x-6\)
  6. Factor the expression completely. View Solution (a) \(x^2+7x+3\)   (b) \(x^2-x-6\)   (c) \(6x^2+x-1\)   (d) \(3x^2-14x+8\)

Solve Quadratic Equations (factoring, completing the square, quadratic formula)

  1. Solve the following equations. View Solution (a) \(1-\dfrac{1}{x^2}=0\)   (b) \(\sqrt{x-1}+3=5\)   (c) \(\sqrt{x}+2=x\)   (d) \(|2x-3|=4\)
  2. Simplify the rational expression. View Solution (a) \(\dfrac{x^{2}+3x+2}{x^{2}-x-2}\)   (b) \(\dfrac{2x^{2}-x-1}{x^{2}-9}\cdot\dfrac{x+3}{2x+1}\)   (c) \(\dfrac{\dfrac{y}{x}-\dfrac{x}{y}}{\dfrac{1}{y}-\dfrac{1}{x}}\)
  3. Simplify each expression. View Solution (a) \(\dfrac{x^3-9x}{2x-6}\)   (b) \(\dfrac{\dfrac{1}{2+h}-\dfrac{1}{2}}{h}\)   (c) \(\dfrac{(5+h)^2-25}{h}\)
  4. Rationalize the numerator. View Solution (a) \(\dfrac{\sqrt{9+h}-3}{h}\)   (b) \(\dfrac{\sqrt{x}-2}{x-4}\)
  5. Let \(f(x)=\sqrt{3x+4}\) and \(g(x)=2x^2-3\). Evaluate the following. View Solution (i) \((f\circ g)(x)\)   (ii) \((f\circ f)(0)\)   (iii) \(g(f(x))\)   (iv) \((g\circ f)(-1)\)
  6. Consider the function \(h(x)\). Find non-identity functions \(f(x)\) and \(g(x)\) such that \(h(x)=f(g(x))\). View Solution (i) \(h(x)=\sqrt[3]{2x+5}\)   (ii) \(h(x)=\dfrac{2}{(3x-7)^2}\)   (iii) \(h(x)=e^{1+x^2}\)
  7. Evaluate the difference quotient as indicated. (a) \(\dfrac{f(a+h)-f(a)}{h}\) for \(f(x)=x^2\) View Solution (b) \(\dfrac{f(4+h)-f(4)}{h}\) for \(f(x)=\sqrt{x}\) View Solution (c) \(\dfrac{f(x+h)-f(x)}{h}\) for \(f(x)=\dfrac{1}{\sqrt{x}}\) View Solution