Rational Exponents Worksheets - Math Monks - Free Printable

Problem: Simplifying Rational Exponents

We are tasked with simplifying expressions involving rational exponents. The goal is to express the final answers using only positive exponents.

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Solution:

#### 1. Simplify \( \left( x^{\frac{1}{2}} y^{\frac{1}{2}} \right)^{-1} \)

Using the property of exponents \((a^m b^n)^p = a^{mp} b^{np}\):
\[
\left( x^{\frac{1}{2}} y^{\frac{1}{2}} \right)^{-1} = x^{\frac{1}{2} \cdot (-1)} y^{\frac{1}{2} \cdot (-1)} = x^{-\frac{1}{2}} y^{-\frac{1}{2}}
\]
To eliminate negative exponents, use \(a^{-n} = \frac{1}{a^n}\):
\[
x^{-\frac{1}{2}} y^{-\frac{1}{2}} = \frac{1}{x^{\frac{1}{2}} y^{\frac{1}{2}}} = \frac{1}{\sqrt{x} \sqrt{y}} = \frac{1}{\sqrt{xy}}
\]

Answer:
\[
\boxed{\frac{1}{\sqrt{xy}}}
\]

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#### 2. Simplify \( (ab^2)^0 \)

Any non-zero number raised to the power of 0 is 1:
\[
(ab^2)^0 = 1
\]

Answer:
\[
\boxed{1}
\]

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#### 3. Simplify \( (81m^6)^{\frac{1}{2}} \)

Using the property \((a \cdot b)^n = a^n \cdot b^n\):
\[
(81m^6)^{\frac{1}{2}} = 81^{\frac{1}{2}} \cdot (m^6)^{\frac{1}{2}}
\]
Simplify each part:
\[
81^{\frac{1}{2}} = \sqrt{81} = 9, \quad (m^6)^{\frac{1}{2}} = m^{6 \cdot \frac{1}{2}} = m^3
\]
Thus:
\[
(81m^6)^{\frac{1}{2}} = 9m^3
\]

Answer:
\[
\boxed{9m^3}
\]

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#### 4. Simplify \( (64n^{12})^{\frac{1}{6}} \)

Using the property \((a \cdot b)^n = a^n \cdot b^n\):
\[
(64n^{12})^{\frac{1}{6}} = 64^{\frac{1}{6}} \cdot (n^{12})^{\frac{1}{6}}
\]
Simplify each part:
\[
64^{\frac{1}{6}} = \sqrt[6]{64} = 2, \quad (n^{12})^{\frac{1}{6}} = n^{12 \cdot \frac{1}{6}} = n^2
\]
Thus:
\[
(64n^{12})^{\frac{1}{6}} = 2n^2
\]

Answer:
\[
\boxed{2n^2}
\]

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#### 5. Simplify \( \frac{(2x)^{\frac{7}{4}}}{(4x)^{\frac{4}{3}}} \)

First, simplify the numerator and denominator separately:
\[
(2x)^{\frac{7}{4}} = 2^{\frac{7}{4}} \cdot x^{\frac{7}{4}}, \quad (4x)^{\frac{4}{3}} = 4^{\frac{4}{3}} \cdot x^{\frac{4}{3}}
\]
Now, divide the two expressions:
\[
\frac{(2x)^{\frac{7}{4}}}{(4x)^{\frac{4}{3}}} = \frac{2^{\frac{7}{4}} \cdot x^{\frac{7}{4}}}{4^{\frac{4}{3}} \cdot x^{\frac{4}{3}}}
\]
Separate the terms:
\[
= \frac{2^{\frac{7}{4}}}{4^{\frac{4}{3}}} \cdot \frac{x^{\frac{7}{4}}}{x^{\frac{4}{3}}}
\]
Simplify the powers of \(x\):
\[
\frac{x^{\frac{7}{4}}}{x^{\frac{4}{3}}} = x^{\frac{7}{4} - \frac{4}{3}} = x^{\frac{21}{12} - \frac{16}{12}} = x^{\frac{5}{12}}
\]
Simplify the powers of 2:
\[
4^{\frac{4}{3}} = (2^2)^{\frac{4}{3}} = 2^{\frac{8}{3}}
\]
Thus:
\[
\frac{2^{\frac{7}{4}}}{4^{\frac{4}{3}}} = \frac{2^{\frac{7}{4}}}{2^{\frac{8}{3}}} = 2^{\frac{7}{4} - \frac{8}{3}} = 2^{\frac{21}{12} - \frac{32}{12}} = 2^{-\frac{11}{12}}
\]
Combine the results:
\[
\frac{(2x)^{\frac{7}{4}}}{(4x)^{\frac{4}{3}}} = 2^{-\frac{11}{12}} \cdot x^{\frac{5}{12}} = \frac{x^{\frac{5}{12}}}{2^{\frac{11}{12}}}
\]

Answer:
\[
\boxed{\frac{x^{\frac{5}{12}}}{2^{\frac{11}{12}}}}
\]

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#### 6. Simplify \( \frac{(x^3 y^2)^{\frac{3}{2}}}{(x^{-1} y^{\frac{2}{3}})^{\frac{1}{4}}} \)

First, simplify the numerator and denominator separately:
\[
(x^3 y^2)^{\frac{3}{2}} = x^{3 \cdot \frac{3}{2}} \cdot y^{2 \cdot \frac{3}{2}} = x^{\frac{9}{2}} \cdot y^3
\]
\[
(x^{-1} y^{\frac{2}{3}})^{\frac{1}{4}} = x^{-1 \cdot \frac{1}{4}} \cdot y^{\frac{2}{3} \cdot \frac{1}{4}} = x^{-\frac{1}{4}} \cdot y^{\frac{1}{6}}
\]
Now, divide the two expressions:
\[
\frac{(x^3 y^2)^{\frac{3}{2}}}{(x^{-1} y^{\frac{2}{3}})^{\frac{1}{4}}} = \frac{x^{\frac{9}{2}} \cdot y^3}{x^{-\frac{1}{4}} \cdot y^{\frac{1}{6}}}
\]
Separate the terms:
\[
= \frac{x^{\frac{9}{2}}}{x^{-\frac{1}{4}}} \cdot \frac{y^3}{y^{\frac{1}{6}}}
\]
Simplify the powers of \(x\) and \(y\):
\[
\frac{x^{\frac{9}{2}}}{x^{-\frac{1}{4}}} = x^{\frac{9}{2} - \left(-\frac{1}{4}\right)} = x^{\frac{9}{2} + \frac{1}{4}} = x^{\frac{18}{4} + \frac{1}{4}} = x^{\frac{19}{4}}
\]
\[
\frac{y^3}{y^{\frac{1}{6}}} = y^{3 - \frac{1}{6}} = y^{\frac{18}{6} - \frac{1}{6}} = y^{\frac{17}{6}}
\]
Combine the results:
\[
\frac{(x^3 y^2)^{\frac{3}{2}}}{(x^{-1} y^{\frac{2}{3}})^{\frac{1}{4}}} = x^{\frac{19}{4}} \cdot y^{\frac{17}{6}}
\]

Answer:
\[
\boxed{x^{\frac{19}{4}} y^{\frac{17}{6}}}
\]

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#### 7. Simplify \( \frac{3x^{\frac{1}{2}} \cdot 3x^{\frac{1}{2}} y^{-\frac{1}{3}}}{3y^{-\frac{7}{4}}} \)

First, simplify the numerator:
\[
3x^{\frac{1}{2}} \cdot 3x^{\frac{1}{2}} y^{-\frac{1}{3}} = (3 \cdot 3) \cdot (x^{\frac{1}{2}} \cdot x^{\frac{1}{2}}) \cdot y^{-\frac{1}{3}} = 9x^{\frac{1}{2} + \frac{1}{2}} y^{-\frac{1}{3}} = 9x^1 y^{-\frac{1}{3}} = 9x y^{-\frac{1}{3}}
\]
Now, divide by the denominator:
\[
\frac{9x y^{-\frac{1}{3}}}{3y^{-\frac{7}{4}}} = \frac{9}{3} \cdot \frac{x}{1} \cdot \frac{y^{-\frac{1}{3}}}{y^{-\frac{7}{4}}} = 3x \cdot y^{-\frac{1}{3} - \left(-\frac{7}{4}\right)}
\]
Simplify the exponent of \(y\):
\[
-\frac{1}{3} - \left(-\frac{7}{4}\right) = -\frac{1}{3} + \frac{7}{4} = -\frac{4}{12} + \frac{21}{12} = \frac{17}{12}
\]
Thus:
\[
\frac{9x y^{-\frac{1}{3}}}{3y^{-\frac{7}{4}}} = 3x y^{\frac{17}{12}}
\]

Answer:
\[
\boxed{3x y^{\frac{17}{12}}}
\]

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#### 8. Simplify \( \frac{2x^{-2} y^{\frac{5}{3}}}{x^{\frac{5}{4}} y^{\frac{5}{3}} \cdot xy^{\frac{1}{2}}} \)

First, simplify the denominator:
\[
x^{\frac{5}{4}} y^{\frac{5}{3}} \cdot xy^{\frac{1}{2}} = x^{\frac{5}{4}} \cdot x^1 \cdot y^{\frac{5}{3}} \cdot y^{\frac{1}{2}} = x^{\frac{5}{4} + 1} \cdot y^{\frac{5}{3} + \frac{1}{2}} = x^{\frac{5}{4} + \frac{4}{4}} \cdot y^{\frac{10}{6} + \frac{3}{6}} = x^{\frac{9}{4}} \cdot y^{\frac{13}{6}}
\]
Now, divide the numerator by the denominator:
\[
\frac{2x^{-2} y^{\frac{5}{3}}}{x^{\frac{9}{4}} y^{\frac{13}{6}}} = 2 \cdot \frac{x^{-2}}{x^{\frac{9}{4}}} \cdot \frac{y^{\frac{5}{3}}}{y^{\frac{13}{6}}}
\]
Simplify the powers of \(x\) and \(y\):
\[
\frac{x^{-2}}{x^{\frac{9}{4}}} = x^{-2 - \frac{9}{4}} = x^{-\frac{8}{4} - \frac{9}{4}} = x^{-\frac{17}{4}}
\]
\[
\frac{y^{\frac{5}{3}}}{y^{\frac{13}{6}}} = y^{\frac{5}{3} - \frac{13}{6}} = y^{\frac{10}{6} - \frac{13}{6}} = y^{-\frac{3}{6}} = y^{-\frac{1}{2}}
\]
Thus:
\[
\frac{2x^{-2} y^{\frac{5}{3}}}{x^{\frac{9}{4}} y^{\frac{13}{6}}} = 2 \cdot x^{-\frac{17}{4}} \cdot y^{-\frac{1}{2}} = \frac{2}{x^{\frac{17}{4}} y^{\frac{1}{2}}}
\]

Answer:
\[
\boxed{\frac{2}{x^{\frac{17}{4}} y^{\frac{1}{2}}}}
\]

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#### 9. Simplify \( \frac{(x^{\frac{4}{3}} y^{\frac{1}{3}} \cdot y)^{-1}}{x^{\frac{1}{3}} y^{-2}} \)

First, simplify the numerator:
\[
x^{\frac{4}{3}} y^{\frac{1}{3}} \cdot y = x^{\frac{4}{3}} \cdot y^{\frac{1}{3} + 1} = x^{\frac{4}{3}} \cdot y^{\frac{1}{3} + \frac{3}{3}} = x^{\frac{4}{3}} \cdot y^{\frac{4}{3}}
\]
Now, take the reciprocal:
\[
\left( x^{\frac{4}{3}} y^{\frac{4}{3}} \right)^{-1} = x^{-\frac{4}{3}} y^{-\frac{4}{3}}
\]
Now, divide by the denominator:
\[
\frac{x^{-\frac{4}{3}} y^{-\frac{4}{3}}}{x^{\frac{1}{3}} y^{-2}} = x^{-\frac{4}{3} - \frac{1}{3}} \cdot y^{-\frac{4}{3} - (-2)} = x^{-\frac{4}{3} - \frac{1}{3}} \cdot y^{-\frac{4}{3} + 2}
\]
Simplify the exponents:
\[
x^{-\frac{4}{3} - \frac{1}{3}} = x^{-\frac{5}{3}}, \quad y^{-\frac{4}{3} + 2} = y^{-\frac{4}{3} + \frac{6}{3}} = y^{\frac{2}{3}}
\]
Thus:
\[
\frac{(x^{\frac{4}{3}} y^{\frac{1}{3}} \cdot y)^{-1}}{x^{\frac{1}{3}} y^{-2}} = x^{-\frac{5}{3}} y^{\frac{2}{3}} = \frac{y^{\frac{2}{3}}}{x^{\frac{5}{3}}}
\]

Answer:
\[
\boxed{\frac{y^{\frac{2}{3}}}{x^{\frac{5}{3}}}}
\]

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#### 10. Simplify \( \left( \frac{x^{\frac{1}{2}} \cdot y^{-2}}{yx^{-\frac{7}{4}}} \right)^4 \)

First, simplify the fraction inside the parentheses:
\[
\frac{x^{\frac{1}{2}} \cdot y^{-2}}{yx^{-\frac{7}{4}}} = \frac{x^{\frac{1}{2}}}{x^{-\frac{7}{4}}} \cdot \frac{y^{-2}}{y}
\]
Simplify the powers of \(x\) and \(y\):
\[
\frac{x^{\frac{1}{2}}}{x^{-\frac{7}{4}}} = x^{\frac{1}{2} - \left(-\frac{7}{4}\right)} = x^{\frac{1}{2} + \frac{7}{4}} = x^{\frac{2}{4} + \frac{7}{4}} = x^{\frac{9}{4}}
\]
\[
\frac{y^{-2}}{y} = y^{-2 - 1} = y^{-3}
\]
Thus:
\[
\frac{x^{\frac{1}{2}} \cdot y^{-2}}{yx^{-\frac{7}{4}}} = x^{\frac{9}{4}} \cdot y^{-3}
\]
Now, raise the entire expression to the power of 4:
\[
\left( x^{\frac{9}{4}} \cdot y^{-3} \right)^4 = \left( x^{\frac{9}{4}} \right)^4 \cdot \left( y^{-3} \right)^4 = x^{\frac{9}{4} \cdot 4} \cdot y^{-3 \cdot 4} = x^9 \cdot y^{-12}
\]
To eliminate the negative exponent:
\[
x^9 \cdot y^{-12} = \frac{x^9}{y^{12}}
\]

Answer:
\[
\boxed{\frac{x^9}{y^{12}}}
\]

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Final Answers:

1. \(\boxed{\frac{1}{\sqrt{xy}}}\)
2. \(\boxed{1}\)
3. \(\boxed{9m^3}\)
4. \(\boxed{2n^2}\)
5. \(\boxed{\frac{x^{\frac{5}{12}}}{2^{\frac{11}{12}}}}\)
6. \(\boxed{x^{\frac{19}{4}} y^{\frac{17}{6}}}\)
7. \(\boxed{3x y^{\frac{17}{12}}}\)
8. \(\boxed{\frac{2}{x^{\frac{17}{4}} y^{\frac{1}{2}}}}\)
9. \(\boxed{\frac{y^{\frac{2}{3}}}{x^{\frac{5}{3}}}}\)
10. \(\boxed{\frac{x^9}{y^{12}}}\)