Roots — Practice Quiz

A Algebra cheat sheet for Roots — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Product Property of Roots

$$\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}$$

Quotient Property of Roots

$$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$

Root of a Root

$$\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$$

Rational Exponents

$$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

Practice quiz

  1. Simplify the expression $\sqrt{72}$.

    • $6\sqrt{2}$
    • $36\sqrt{2}$
    • $2\sqrt{6}$
    • $6\sqrt{12}$

    Answer: $6\sqrt{2}$

  2. Simplify the expression $\sqrt{\frac{49}{16}}$.

    • $\frac{7}{4}$
    • $\frac{7}{16}$
    • $\frac{49}{4}$
    • $\frac{1}{4}$

    Answer: $\frac{7}{4}$

  3. Evaluate $\sqrt[3]{\sqrt{64}}$.

    • $2$
    • $4$
    • $8$
    • $16$

    Answer: $2$

  4. Write $x^{3/5}$ in radical form.

    • $\sqrt[3]{x^5}$
    • $\sqrt[5]{x^3}$
    • $(\sqrt{x})^3$
    • $\frac{3}{5}x$

    Answer: $\sqrt[5]{x^3}$

  5. Write $(\sqrt[4]{y})^7$ in rational exponent form.

    • $y^{4/7}$
    • $y^{7/4}$
    • $y^{1/28}$
    • $y^{7-4}$

    Answer: $y^{7/4}$

  6. Simplify $(8x^6)^{2/3}$.

    • $4x^4$
    • $2x^4$
    • $4x^2$
    • $2x^2$

    Answer: $4x^4$

  7. Simplify $\sqrt{\frac{18x^5}{2x}}$ for $x > 0$.

    • $3x^2$
    • $9x^2$
    • $3x\sqrt{x}$
    • $9x\sqrt{x}$

    Answer: $3x^2$

  8. Simplify $\sqrt[4]{\sqrt{x^{16}}}$.

    • $x^2$
    • $x^4$
    • $x^8$
    • $x^{16}$

    Answer: $x^2$

  9. Evaluate $(27^{1/3})^2$.

    • $3$
    • $9$
    • $27$
    • $81$

    Answer: $9$

  10. Which of the following is equivalent to $\frac{\sqrt[3]{16x^7}}{\sqrt[3]{2x}}$?

    • $2x^2$
    • $4x^2$
    • $2x\sqrt[3]{x}$
    • $4x\sqrt[3]{x}$

    Answer: $2x^2$

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