Worksheet: Exponents - Division Properties of Exponents - Rules ... - Free Printable

Let’s solve each expression step by step using the rules of exponents. We’ll simplify each one carefully.

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1) $\frac{x^5}{x^2}$

When dividing powers with the same base, subtract the exponents:
$x^{5-2} = x^3$

✔ Final: $x^3$

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2) $\frac{a^8}{a^4}$

Same rule: subtract exponents → $a^{8-4} = a^4$

✔ Final: $a^4$

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3) $\frac{10y^7}{5y^3}$

First, divide coefficients: $10 ÷ 5 = 2$
Then, subtract exponents for y: $y^{7-3} = y^4$
So: $2y^4$

✔ Final: $2y^4$

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4) $\frac{p^9}{p^9}$

Subtract exponents: $p^{9-9} = p^0 = 1$ (any nonzero number to power 0 is 1)

✔ Final: $1$

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5) $\left(\frac{b^2}{c}\right)^3$

Apply exponent to numerator and denominator:
Numerator: $(b^2)^3 = b^{6}$
Denominator: $c^3$
So: $\frac{b^6}{c^3}$

✔ Final: $\frac{b^6}{c^3}$

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6) $\frac{z^0}{z^4}$

$z^0 = 1$, so this becomes $\frac{1}{z^4}$

Alternatively, subtract exponents: $z^{0-4} = z^{-4} = \frac{1}{z^4}$

✔ Final: $\frac{1}{z^4}$

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7) $\frac{-m^3 n^2}{m n}$

Break it down:

Coefficients: -1 stays
For m: $m^{3-1} = m^2$
For n: $n^{2-1} = n^1 = n$
So: $-m^2 n$

✔ Final: $-m^2 n$

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8) $\frac{(2x^2 y)(3x^4 y^2)}{6x^5 y^3}$

First, multiply numerator:

$(2x^2 y)(3x^4 y^2) = 2×3 × x^{2+4} × y^{1+2} = 6x^6 y^3$

Now divide by denominator: $\frac{6x^6 y^3}{6x^5 y^3}$

Cancel 6/6 = 1
$x^{6-5} = x^1 = x$
$y^{3-3} = y^0 = 1$

So: $x × 1 = x$

✔ Final: $x$

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9) $\frac{(-3a^2 b)(4ab^3)}{-12a^3 b^4}$

Multiply numerator first:

$(-3)(4) = -12$
$a^{2+1} = a^3$
$b^{1+3} = b^4$
So numerator: $-12a^3 b^4$

Denominator: $-12a^3 b^4$

So: $\frac{-12a^3 b^4}{-12a^3 b^4} = 1$

(Negative divided by negative is positive; everything else cancels.)

✔ Final: $1$

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10) $\frac{x^7}{x^9}$

Subtract exponents: $x^{7-9} = x^{-2} = \frac{1}{x^2}$

✔ Final: $\frac{1}{x^2}$

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11) $\frac{y^5}{y^8}$

$y^{5-8} = y^{-3} = \frac{1}{y^3}$

✔ Final: $\frac{1}{y^3}$

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12) $\frac{10k^6}{5k^9}$

Divide coefficients: $10 ÷ 5 = 2$
Exponents: $k^{6-9} = k^{-3} = \frac{1}{k^3}$
So: $2 × \frac{1}{k^3} = \frac{2}{k^3}$

✔ Final: $\frac{2}{k^3}$

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Final Answer:
1) $x^3$
2) $a^4$
3) $2y^4$
4) $1$
5) $\frac{b^6}{c^3}$
6) $\frac{1}{z^4}$
7) $-m^2 n$
8) $x$
9) $1$
10) $\frac{1}{x^2}$
11) $\frac{1}{y^3}$
12) $\frac{2}{k^3}$