Roots — Hard Practice Quiz

A Algebra cheat sheet for Roots — every key formula with its symbols defined — plus a hard-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 $\sqrt[3]{x^2 \sqrt{x^5}}$.

    • $x^{3/2}$
    • $x^{7/6}$
    • $x^{19/6}$
    • $x^{1/2}$

    Answer: $x^{3/2}$

  2. If $y = \sqrt[4]{x^3}$ and $z = \sqrt{x}$, express $y$ in terms of $z$.

    • $z^{3/2}$
    • $z^{3/4}$
    • $z^6$
    • $z^{1/2}$

    Answer: $z^{3/2}$

  3. Simplify the expression $\frac{\sqrt[3]{a^5 b^7}}{\sqrt{a b^2}}$. Assume $a, b > 0$.

    • $a^{7/6} b^{4/3}$
    • $a^{13/6} b^{13/3}$
    • $a^{7/6} b^{1/3}$
    • $a^{1/6} b^{4/3}$

    Answer: $a^{7/6} b^{4/3}$

  4. Given $P = \sqrt[n]{x^m}$ and $Q = \sqrt[m]{x^n}$, what is the value of $\frac{P}{Q}$ in terms of $x, m, n$?

    • $x^{\frac{m^2 - n^2}{mn}}$
    • $x^{\frac{m-n}{mn}}$
    • $x^{\frac{m^2+n^2}{mn}}$
    • $x^{\frac{n^2-m^2}{mn}}$

    Answer: $x^{\frac{m^2 - n^2}{mn}}$

  5. If $A = \sqrt[3]{\sqrt{x}}$ and $B = \sqrt{x^3}$, find the value of $(A \cdot B)^2$.

    • $x^{10/3}$
    • $x^{5/6}$
    • $x^{1/3}$
    • $x^{5/3}$

    Answer: $x^{10/3}$

  6. Simplify $\sqrt[4]{\frac{16x^8 y^{12}}{z^{16}}}$. Assume all variables are positive.

    • $\frac{2 x^2 y^3}{z^4}$
    • $\frac{4 x^2 y^3}{z^4}$
    • $\frac{2 x^4 y^8}{z^{12}}$
    • $\frac{2 x^2 y^3}{z^8}$

    Answer: $\frac{2 x^2 y^3}{z^4}$

  7. If $k = \sqrt[3]{m^2}$ and $m = \sqrt{p^3}$, express $k$ in terms of $p$.

    • $p$
    • $p^{4/9}$
    • $p^{9/4}$
    • $p^{1/2}$

    Answer: $p$

  8. Which of the following expressions is equivalent to $\sqrt[3]{x^2 \sqrt[4]{x^3}}$?

    • $x^{11/12}$
    • $\sqrt[7]{x^5}$
    • $x^{5/12}$
    • $\sqrt[12]{x^8}$

    Answer: $x^{11/12}$

  9. If $a = \sqrt[x]{y^z}$ and $b = \sqrt[z]{y^x}$, what is the value of $(a/b)^{xz}$?

    • $y^{z^2 - x^2}$
    • $y^{x^2 - z^2}$
    • $y^{z^2 + x^2}$
    • $y^{1/xz}$

    Answer: $y^{z^2 - x^2}$

  10. Simplify $\sqrt[n]{(x^n y^m)^2 \cdot \sqrt[m]{x^{mn}}}$. Assume $x, y > 0$.

    • $x^3 y^{2m/n}$
    • $x^3 y^{2m}$
    • $x^3 y^{m/n}$
    • $x^2 y^{2m/n}$

    Answer: $x^3 y^{2m/n}$

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