Unit 8 · Checkpoint quiz

Unit 8 checkpoint: Radicals & complex numbers

12 questions · about 25-30 minutes · no time limit

Answer every question, then press Grade my answers at the bottom. You'll see what you got right, a worked solution for each question, and links to the lessons worth revisiting. Pencil and paper are encouraged.

This is practice, not a test of you. A low score just shows where to look next.

Simplify completely: \(\sqrt{48x^5}\). Assume \(x \ge 0\).

  1. \(2x^2\sqrt{12x}\)
  2. \(16x^2\sqrt{3x}\)
  3. \(4x^2\sqrt{3x}\)
  4. \(4x^4\sqrt{3x}\)

Split into a perfect-square part and the rest: \(48 = 16 \cdot 3\) and \(x^5 = x^4 \cdot x\).

\(\sqrt{16x^4} = 4x^2\) comes out; \(3x\) stays inside.

Answer: \(4x^2\sqrt{3x}\). (Choice a is equal but not finished: \(12 = 4 \cdot 3\) still has a perfect square inside.)

Evaluate \(\sqrt[3]{-64} + \sqrt{81}\).

\((-4)^3 = -64\), so \(\sqrt[3]{-64} = -4\). (A cube root of a negative number is negative.)

\(9^2 = 81\), so \(\sqrt{81} = 9\).

\(-4 + 9 = 5\). Answer: \(5\).

Simplify \(x^{2/3} \cdot x^{1/4}\). Write your answer as one power of \(x\).

Same base, multiplying: add the exponents.

\(\dfrac{2}{3} + \dfrac{1}{4} = \dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}\).

Answer: \(x^{11/12}\).

Evaluate \(\left(\dfrac{8}{27}\right)^{-2/3}\).

The negative exponent flips the fraction: \(\left(\dfrac{27}{8}\right)^{2/3}\).

The denominator 3 means cube root first: \(\sqrt[3]{\dfrac{27}{8}} = \dfrac{3}{2}\).

Then square: \(\left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\). Answer: \(\dfrac{9}{4}\).

Simplify \(3\sqrt{12} - \sqrt{27} + \sqrt{75}\).

  1. \(4\sqrt{3}\)
  2. \(8\sqrt{3}\)
  3. \(3\sqrt{60}\)
  4. \(14\sqrt{3}\)

Simplify each radical first: \(\sqrt{12} = 2\sqrt{3}\), \(\sqrt{27} = 3\sqrt{3}\), \(\sqrt{75} = 5\sqrt{3}\).

So the expression is \(3 \cdot 2\sqrt{3} - 3\sqrt{3} + 5\sqrt{3} = 6\sqrt{3} - 3\sqrt{3} + 5\sqrt{3}\).

Now they are like radicals: \(6 - 3 + 5 = 8\). Answer: \(8\sqrt{3}\).

Simplify \(\dfrac{\sqrt{72x^3}}{\sqrt{2x}}\). Assume \(x > 0\).

Put it under one radical: \(\sqrt{\dfrac{72x^3}{2x}}\).

Divide inside: \(\dfrac{72x^3}{2x} = 36x^2\).

\(\sqrt{36x^2} = 6x\). Answer: \(6x\).

Rationalize the denominator and simplify: \(\dfrac{2}{\sqrt{7} + \sqrt{3}}\).

  1. \(\dfrac{\sqrt{7} + \sqrt{3}}{2}\)
  2. \(\dfrac{\sqrt{7} - \sqrt{3}}{5}\)
  3. \(\dfrac{\sqrt{7} - \sqrt{3}}{4}\)
  4. \(\dfrac{\sqrt{7} - \sqrt{3}}{2}\)

Multiply top and bottom by the conjugate \(\sqrt{7} - \sqrt{3}\).

Bottom: \((\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 7 - 3 = 4\). Top: \(2(\sqrt{7} - \sqrt{3})\).

\(\dfrac{2(\sqrt{7} - \sqrt{3})}{4} = \dfrac{\sqrt{7} - \sqrt{3}}{2}\). Answer: \(\dfrac{\sqrt{7} - \sqrt{3}}{2}\).

Solve \(\sqrt{2x + 7} = x + 2\). Check for extraneous solutions.

Square both sides: \(2x + 7 = (x + 2)^2 = x^2 + 4x + 4\).

Standard form: \(0 = x^2 + 2x - 3 = (x + 3)(x - 1)\), so \(x = -3\) or \(x = 1\).

Check \(x = 1\): \(\sqrt{9} = 3\) and \(1 + 2 = 3\). It works.

Check \(x = -3\): \(\sqrt{1} = 1\) but \(-3 + 2 = -1\). Extraneous. Answer: \(x = 1\) only.

Multiply and write in the form \(a + bi\): \((3 - 2i)(4 + i)\).

FOIL: \(12 + 3i - 8i - 2i^2\).

Replace \(i^2\) with \(-1\): \(-2i^2 = +2\).

\(12 + 2 + (3 - 8)i = 14 - 5i\). Answer: \(14 - 5i\).

Divide and write in the form \(a + bi\): \(\dfrac{5 + i}{1 - i}\).

Multiply top and bottom by the conjugate of the bottom, \(1 + i\).

Bottom: \((1 - i)(1 + i) = 1 - i^2 = 1 + 1 = 2\).

Top: \((5 + i)(1 + i) = 5 + 5i + i + i^2 = 5 + 6i - 1 = 4 + 6i\).

\(\dfrac{4 + 6i}{2} = 2 + 3i\). Answer: \(2 + 3i\).

Factor: \(x^2 - 2xy - 15y^2\).

Look for two numbers that multiply to \(-15\) and add to \(-2\): they are \(-5\) and \(3\).

Each \(y\) goes with its number: \((x - 5y)(x + 3y)\).

Check: \(x^2 + 3xy - 5xy - 15y^2 = x^2 - 2xy - 15y^2\). Answer: \((x - 5y)(x + 3y)\).

Simplify \(\dfrac{x^2 - 9}{x^2 + x - 6}\).

Factor the top: \(x^2 - 9 = (x - 3)(x + 3)\).

Factor the bottom: \(x^2 + x - 6 = (x + 3)(x - 2)\).

Cancel the common factor \((x + 3)\). Answer: \(\dfrac{x - 3}{x - 2}\).

Want to look something up first? Radicals and complex numbers cheat sheet · Glossary

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