Exponents Worksheets - Free Printable

To solve the given problems using the laws of exponents, we will simplify each expression step by step. Here are the solutions:

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1. \(\left( \frac{x^3}{y^2} \right)^4\)

Using the power of a quotient rule: \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\),

\[
\left( \frac{x^3}{y^2} \right)^4 = \frac{(x^3)^4}{(y^2)^4}
\]

Using the power of a power rule: \((a^m)^n = a^{m \cdot n}\),

\[
(x^3)^4 = x^{3 \cdot 4} = x^{12}, \quad (y^2)^4 = y^{2 \cdot 4} = y^8
\]

Thus,

\[
\left( \frac{x^3}{y^2} \right)^4 = \frac{x^{12}}{y^8}
\]

Answer: \(\boxed{\frac{x^{12}}{y^8}}\)

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2. \((a^5 b^3)^4 (ab^2)^3\)

First, apply the power of a product rule: \((ab)^n = a^n b^n\),

\[
(a^5 b^3)^4 = a^{5 \cdot 4} b^{3 \cdot 4} = a^{20} b^{12}
\]
\[
(ab^2)^3 = a^{1 \cdot 3} b^{2 \cdot 3} = a^3 b^6
\]

Now multiply the results:

\[
(a^5 b^3)^4 (ab^2)^3 = a^{20} b^{12} \cdot a^3 b^6
\]

Using the product of powers rule: \(a^m \cdot a^n = a^{m+n}\),

\[
a^{20} \cdot a^3 = a^{20+3} = a^{23}, \quad b^{12} \cdot b^6 = b^{12+6} = b^{18}
\]

Thus,

\[
(a^5 b^3)^4 (ab^2)^3 = a^{23} b^{18}
\]

Answer: \(\boxed{a^{23} b^{18}}\)

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3. \(\left( \frac{3m^2 n^3}{2mn^4} \right)^{-2}\)

First, simplify the fraction inside the parentheses:

\[
\frac{3m^2 n^3}{2mn^4} = \frac{3m^{2-1} n^{3-4}}{2} = \frac{3m^1 n^{-1}}{2} = \frac{3m}{2n}
\]

Now apply the negative exponent rule: \(a^{-n} = \frac{1}{a^n}\),

\[
\left( \frac{3m}{2n} \right)^{-2} = \left( \frac{2n}{3m} \right)^2
\]

Using the power of a quotient rule:

\[
\left( \frac{2n}{3m} \right)^2 = \frac{(2n)^2}{(3m)^2} = \frac{2^2 n^2}{3^2 m^2} = \frac{4n^2}{9m^2}
\]

Answer: \(\boxed{\frac{4n^2}{9m^2}}\)

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4. \((5p^3 q^2)(2p^4 q^3)^2\)

First, simplify \((2p^4 q^3)^2\):

\[
(2p^4 q^3)^2 = 2^2 (p^4)^2 (q^3)^2 = 4 p^{4 \cdot 2} q^{3 \cdot 2} = 4 p^8 q^6
\]

Now multiply the results:

\[
(5p^3 q^2)(4 p^8 q^6) = 5 \cdot 4 \cdot p^3 \cdot p^8 \cdot q^2 \cdot q^6 = 20 p^{3+8} q^{2+6} = 20 p^{11} q^8
\]

Answer: \(\boxed{20p^{11}q^8}\)

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5. \(\frac{(8a^5 b^3)(2b^4)}{4a^{-6}}\)

First, simplify the numerator:

\[
(8a^5 b^3)(2b^4) = 8 \cdot 2 \cdot a^5 \cdot b^3 \cdot b^4 = 16 a^5 b^{3+4} = 16 a^5 b^7
\]

Now divide by the denominator:

\[
\frac{16 a^5 b^7}{4a^{-6}} = \frac{16}{4} \cdot \frac{a^5}{a^{-6}} \cdot b^7 = 4 \cdot a^{5 - (-6)} \cdot b^7 = 4 \cdot a^{5+6} \cdot b^7 = 4 a^{11} b^7
\]

Answer: \(\boxed{4a^{11}b^7}\)

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6. \((x^{-2} y^3)^{-4}(x^3 y^{-4})^{-4}\)

First, simplify each term using the power of a power rule:

\[
(x^{-2} y^3)^{-4} = x^{-2 \cdot -4} y^{3 \cdot -4} = x^8 y^{-12}
\]
\[
(x^3 y^{-4})^{-4} = x^{3 \cdot -4} y^{-4 \cdot -4} = x^{-12} y^{16}
\]

Now multiply the results:

\[
(x^{-2} y^3)^{-4}(x^3 y^{-4})^{-4} = x^8 y^{-12} \cdot x^{-12} y^{16}
\]

Using the product of powers rule:

\[
x^8 \cdot x^{-12} = x^{8 + (-12)} = x^{-4}, \quad y^{-12} \cdot y^{16} = y^{-12 + 16} = y^4
\]

Thus,

\[
(x^{-2} y^3)^{-4}(x^3 y^{-4})^{-4} = x^{-4} y^4
\]

Using the negative exponent rule:

\[
x^{-4} y^4 = \frac{y^4}{x^4}
\]

Answer: \(\boxed{\frac{y^4}{x^4}}\)

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7. \(\left( \frac{6m^2 n^3}{3t^2} \right)^2\)

First, simplify the fraction inside the parentheses:

\[
\frac{6m^2 n^3}{3t^2} = \frac{6}{3} \cdot \frac{m^2 n^3}{t^2} = 2 \cdot \frac{m^2 n^3}{t^2} = \frac{2m^2 n^3}{t^2}
\]

Now apply the power of a quotient rule:

\[
\left( \frac{2m^2 n^3}{t^2} \right)^2 = \frac{(2m^2 n^3)^2}{(t^2)^2}
\]

Simplify the numerator and denominator:

\[
(2m^2 n^3)^2 = 2^2 (m^2)^2 (n^3)^2 = 4 m^{2 \cdot 2} n^{3 \cdot 2} = 4 m^4 n^6
\]
\[
(t^2)^2 = t^{2 \cdot 2} = t^4
\]

Thus,

\[
\left( \frac{6m^2 n^3}{3t^2} \right)^2 = \frac{4m^4 n^6}{t^4}
\]

Answer: \(\boxed{\frac{4m^4n^6}{t^4}}\)

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8. \(\frac{(2x^{-3} y^4)(3x^2 y^{-3})}{(x^4 y^{-2})}\)

First, simplify the numerator:

\[
(2x^{-3} y^4)(3x^2 y^{-3}) = 2 \cdot 3 \cdot x^{-3} \cdot x^2 \cdot y^4 \cdot y^{-3} = 6 x^{-3+2} y^{4+(-3)} = 6 x^{-1} y^1 = \frac{6y}{x}
\]

Now divide by the denominator:

\[
\frac{\frac{6y}{x}}{x^4 y^{-2}} = \frac{6y}{x} \cdot \frac{1}{x^4 y^{-2}} = \frac{6y \cdot y^2}{x \cdot x^4} = \frac{6y^{1+2}}{x^{1+4}} = \frac{6y^3}{x^5}
\]

Answer: \(\boxed{\frac{6y^3}{x^5}}\)

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9. \((u^{-2} v^3)(uv^{-4})(u^3 v^{-2})\)

Multiply the terms together:

\[
(u^{-2} v^3)(uv^{-4})(u^3 v^{-2}) = u^{-2} \cdot u^1 \cdot u^3 \cdot v^3 \cdot v^{-4} \cdot v^{-2}
\]

Using the product of powers rule:

\[
u^{-2+1+3} = u^{2}, \quad v^{3+(-4)+(-2)} = v^{-3}
\]

Thus,

\[
(u^{-2} v^3)(uv^{-4})(u^3 v^{-2}) = u^2 v^{-3}
\]

Using the negative exponent rule:

\[
u^2 v^{-3} = \frac{u^2}{v^3}
\]

Answer: \(\boxed{\frac{u^2}{v^3}}\)

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10. \(\frac{8p^3 q^{-6}}{(2p^{-3} q^2)(p^2 q^{-4})}\)

First, simplify the denominator:

\[
(2p^{-3} q^2)(p^2 q^{-4}) = 2 \cdot p^{-3} \cdot p^2 \cdot q^2 \cdot q^{-4} = 2 \cdot p^{-3+2} \cdot q^{2+(-4)} = 2 \cdot p^{-1} \cdot q^{-2} = \frac{2q^{-2}}{p}
\]

Now divide the numerator by the denominator:

\[
\frac{8p^3 q^{-6}}{\frac{2q^{-2}}{p}} = 8p^3 q^{-6} \cdot \frac{p}{2q^{-2}} = \frac{8p^3 \cdot p \cdot q^{-6}}{2q^{-2}} = \frac{8p^{3+1} q^{-6}}{2q^{-2}} = \frac{8p^4 q^{-6}}{2q^{-2}}
\]

Simplify the fraction:

\[
\frac{8}{2} \cdot \frac{p^4}{1} \cdot \frac{q^{-6}}{q^{-2}} = 4 p^4 q^{-6-(-2)} = 4 p^4 q^{-4}
\]

Using the negative exponent rule:

\[
4 p^4 q^{-4} = \frac{4p^4}{q^4}
\]

Answer: \(\boxed{\frac{4p^4}{q^4}}\)

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11. \(\left( \frac{3x^{-1} y^2}{z^3} \right)^{-4}\)

Using the negative exponent rule:

\[
\left( \frac{3x^{-1} y^2}{z^3} \right)^{-4} = \left( \frac{z^3}{3x^{-1} y^2} \right)^4
\]

Using the power of a quotient rule:

\[
\left( \frac{z^3}{3x^{-1} y^2} \right)^4 = \frac{(z^3)^4}{(3x^{-1} y^2)^4}
\]

Simplify the numerator and denominator:

\[
(z^3)^4 = z^{3 \cdot 4} = z^{12}
\]
\[
(3x^{-1} y^2)^4 = 3^4 (x^{-1})^4 (y^2)^4 = 81 x^{-4} y^8
\]

Thus,

\[
\left( \frac{3x^{-1} y^2}{z^3} \right)^{-4} = \frac{z^{12}}{81 x^{-4} y^8}
\]

Using the negative exponent rule:

\[
\frac{z^{12}}{81 x^{-4} y^8} = \frac{z^{12} x^4}{81 y^8}
\]

Answer: \(\boxed{\frac{z^{12}x^4}{81y^8}}\)

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12. \((3m^{-2} n^3)(2mn^2)(m^3 n^{-4})\)

Multiply the terms together:

\[
(3m^{-2} n^3)(2mn^2)(m^3 n^{-4}) = 3 \cdot 2 \cdot m^{-2} \cdot m^1 \cdot m^3 \cdot n^3 \cdot n^2 \cdot n^{-4}
\]

Using the product of powers rule:

\[
3 \cdot 2 = 6, \quad m^{-2+1+3} = m^2, \quad n^{3+2+(-4)} = n^1
\]

Thus,

\[
(3m^{-2} n^3)(2mn^2)(m^3 n^{-4}) = 6m^2 n
\]

Answer: \(\boxed{6m^2n}\)

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13. \((4w^2)^3(w^{-3})^4(w^{-4})^2\)

First, simplify each term using the power of a power rule:

\[
(4w^2)^3 = 4^3 (w^2)^3 = 64 w^{2 \cdot 3} = 64 w^6
\]
\[
(w^{-3})^4 = w^{-3 \cdot 4} = w^{-12}
\]
\[
(w^{-4})^2 = w^{-4 \cdot 2} = w^{-8}
\]

Now multiply the results:

\[
(4w^2)^3(w^{-3})^4(w^{-4})^2 = 64 w^6 \cdot w^{-12} \cdot w^{-8}
\]

Using the product of powers rule:

\[
w^6 \cdot w^{-12} \cdot w^{-8} = w^{6 + (-12) + (-8)} = w^{-14}
\]

Thus,

\[
(4w^2)^3(w^{-3})^4(w^{-4})^2 = 64 w^{-14}
\]

Using the negative exponent rule:

\[
64 w^{-14} = \frac{64}{w^{14}}
\]

Answer: \(\boxed{\frac{64}{w^{14}}}\)

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14. \(\left( \frac{4x^2 y^3}{2xy^2} \right)^{-3}\)

First, simplify the fraction inside the parentheses:

\[
\frac{4x^2 y^3}{2xy^2} = \frac{4}{2} \cdot \frac{x^2}{x} \cdot \frac{y^3}{y^2} = 2 \cdot x^{2-1} \cdot y^{3-2} = 2xy
\]

Now apply the negative exponent rule:

\[
\left( 2xy \right)^{-3} = \frac{1}{(2xy)^3}
\]

Using the power of a product rule:

\[
(2xy)^3 = 2^3 x^3 y^3 = 8x^3 y^3
\]

Thus,

\[
\left( \frac{4x^2 y^3}{2xy^2} \right)^{-3} = \frac{1}{8x^3 y^3}
\]

Answer: \(\boxed{\frac{1}{8x^3y^3}}\)

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15. \(\frac{(2a^3 b^2)(4b^3)}{4a^{-2}}\)

First, simplify the numerator:

\[
(2a^3 b^2)(4b^3) = 2 \cdot 4 \cdot a^3 \cdot b^2 \cdot b^3 = 8 a^3 b^{2+3} = 8 a^3 b^5
\]

Now divide by the denominator:

\[
\frac{8 a^3 b^5}{4a^{-2}} = \frac{8}{4} \cdot \frac{a^3}{a^{-2}} \cdot b^5 = 2 \cdot a^{3 - (-2)} \cdot b^5 = 2 \cdot a^{3+2} \cdot b^5 = 2 a^5 b^5
\]

Answer: \(\boxed{2a^5b^5}\)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & \frac{x^{12}}{y^8} \\
2. & a^{23} b^{18} \\
3. & \frac{4n^2}{9m^2} \\
4. & 20p^{11}q^8 \\
5. & 4a^{11}b^7 \\
6. & \frac{y^4}{x^4} \\
7. & \frac{4m^4n^6}{t^4} \\
8. & \frac{6y^3}{x^5} \\
9. & \frac{u^2}{v^3} \\
10. & \frac{4p^4}{q^4} \\
11. & \frac{z^{12}x^4}{81y^8} \\
12. & 6m^2n \\
13. & \frac{64}{w^{14}} \\
14. & \frac{1}{8x^3y^3} \\
15. & 2a^5b^5 \\
\end{array}
}
\]