Powers — Hard Practice Quiz

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

Formulas & key concepts

Product Rule

$$a^n \cdot a^m = a^{n+m}$$

Quotient Rule

$$\frac{a^n}{a^m} = a^{n-m}$$

Power of a Power

$$(a^n)^m = a^{nm}$$

Power of a Product

$$(ab)^n = a^n b^n$$

Power of a Quotient

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Negative Exponent

$$a^{-n} = \frac{1}{a^n}$$

Zero Exponent (\(a \neq 0\))

$$a^0 = 1$$

Practice quiz

  1. Simplify the expression: $(\frac{x^3 y^{-2}}{x^{-1} y^4})^2$

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

    Answer: $\frac{x^8}{y^{12}}$

  2. If $(2^{3x} \cdot 2^{-x})^2 = 2^{12}$, what is the value of $x$?

    • $2$
    • $3$
    • $4$
    • $6$

    Answer: $3$

  3. Given $A = (x^a)^b$ and $B = x^{a+b}$. If $x > 1$, under what condition is $A > B$?

    • $a=b$
    • $a+b > ab$
    • $(a-1)(b-1) > 1$
    • $a < b$

    Answer: $(a-1)(b-1) > 1$

  4. Simplify the expression: $\frac{(p^2 q^{-3})^0 \cdot (p^{-1} q^2)^3}{p^4 q^{-5}}$ (Assume $p, q \neq 0$)

    • $\frac{q^{11}}{p^7}$
    • $\frac{p^7}{q^{11}}$
    • $p^7 q^{11}$
    • $1$

    Answer: $\frac{q^{11}}{p^7}$

  5. Given $A = x^{2m}$ and $B = x^{m+1}$, express $A$ in terms of $B$ and $x$.

    • $B^2 x^{-2}$
    • $B^2 x^2$
    • $B x^{-1}$
    • $B^2$

    Answer: $B^2 x^{-2}$

  6. If $x = 2^a$ and $y = 2^{-a}$, what is the value of $\frac{x^2}{y^{-1}}$?

    • $x$
    • $y$
    • $x^2$
    • $y^2$

    Answer: $x$

  7. Simplify: $(\frac{a^2 b^{-3} c^0}{a^{-1} b^2})^{-2} \cdot (a^3 b^{-1})^2$ (Assume $a, b, c \neq 0$)

    • $b^8$
    • $a^{12} b^{-12}$
    • $\frac{1}{b^8}$
    • $a^6 b^8$

    Answer: $b^8$

  8. If $(\frac{x^2}{y^{-1}})^3 = \frac{x^6}{8}$ and $x, y \neq 0$, what is the value of $y$?

    • $2$
    • $\frac{1}{2}$
    • $4$
    • $\frac{1}{4}$

    Answer: $\frac{1}{2}$

  9. Consider the expression $E = (x^a)^b$. If $x > 1$, how does $E$ change if $a$ is doubled and $b$ is halved?

    • $E$ is doubled
    • $E$ is halved
    • $E$ remains unchanged
    • $E$ is squared

    Answer: $E$ remains unchanged

  10. Simplify: $\frac{(m^{-2} n^3)^{-1}}{(m^3 n^{-1})^2} \cdot (\frac{m}{n})^{-3}$ (Assume $m, n \neq 0$)

    • $\frac{n^2}{m^7}$
    • $\frac{m^7}{n^2}$
    • $m^7 n^2$
    • $m^{-1} n^{-4}$

    Answer: $\frac{n^2}{m^7}$

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