SOLUTION: Kuta Software Infinite Algebra 2: Simplifying Rational ... - Free Printable

The task involves simplifying expressions with rational exponents. Let's solve each problem step by step.

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Problem 1: Simplify \((n^4)^{\frac{3}{2}}\)

#### Solution:
1. Use the power rule of exponents: \((a^m)^n = a^{m \cdot n}\).
\[
(n^4)^{\frac{3}{2}} = n^{4 \cdot \frac{3}{2}}
\]
2. Simplify the exponent:
\[
4 \cdot \frac{3}{2} = \frac{4 \cdot 3}{2} = \frac{12}{2} = 6
\]
3. Therefore:
\[
(n^4)^{\frac{3}{2}} = n^6
\]

#### Final Answer:
\[
\boxed{n^6}
\]

---

Problem 2: Simplify \((27p^6)^{\frac{5}{3}}\)

#### Solution:
1. Use the power rule of exponents for both the base \(27\) and the variable \(p^6\):
\[
(27p^6)^{\frac{5}{3}} = 27^{\frac{5}{3}} \cdot (p^6)^{\frac{5}{3}}
\]
2. Simplify \(27^{\frac{5}{3}}\):
- Note that \(27 = 3^3\). So, \(27^{\frac{5}{3}} = (3^3)^{\frac{5}{3}}\).
- Apply the power rule: \((a^m)^n = a^{m \cdot n}\).
\[
(3^3)^{\frac{5}{3}} = 3^{3 \cdot \frac{5}{3}} = 3^5
\]
- Calculate \(3^5\):
\[
3^5 = 243
\]
3. Simplify \((p^6)^{\frac{5}{3}}\):
- Apply the power rule: \((a^m)^n = a^{m \cdot n}\).
\[
(p^6)^{\frac{5}{3}} = p^{6 \cdot \frac{5}{3}} = p^{\frac{30}{3}} = p^{10}
\]
4. Combine the results:
\[
(27p^6)^{\frac{5}{3}} = 243 \cdot p^{10} = 243p^{10}
\]

#### Final Answer:
\[
\boxed{243p^{10}}
\]

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Problem 3: Simplify \(\left(\frac{x^{-2}y^3}{x^4y^{-1}}\right)^{-\frac{1}{2}}\)

#### Solution:
1. Simplify the fraction inside the parentheses:
\[
\frac{x^{-2}y^3}{x^4y^{-1}} = x^{-2-4} \cdot y^{3-(-1)} = x^{-6} \cdot y^{3+1} = x^{-6}y^4
\]
2. Rewrite the expression with the simplified fraction:
\[
\left(x^{-6}y^4\right)^{-\frac{1}{2}}
\]
3. Apply the power rule of exponents to both \(x^{-6}\) and \(y^4\):
\[
\left(x^{-6}y^4\right)^{-\frac{1}{2}} = x^{-6 \cdot -\frac{1}{2}} \cdot y^{4 \cdot -\frac{1}{2}}
\]
4. Simplify the exponents:
- For \(x\):
\[
-6 \cdot -\frac{1}{2} = \frac{6}{2} = 3
\]
- For \(y\):
\[
4 \cdot -\frac{1}{2} = -\frac{4}{2} = -2
\]
5. Combine the results:
\[
x^3 \cdot y^{-2} = \frac{x^3}{y^2}
\]

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

---

Problem 4: Simplify \(\left(\frac{a^{-3}b^2}{a^2b^{-4}}\right)^{-\frac{2}{3}}\)

#### Solution:
1. Simplify the fraction inside the parentheses:
\[
\frac{a^{-3}b^2}{a^2b^{-4}} = a^{-3-2} \cdot b^{2-(-4)} = a^{-5} \cdot b^{2+4} = a^{-5}b^6
\]
2. Rewrite the expression with the simplified fraction:
\[
\left(a^{-5}b^6\right)^{-\frac{2}{3}}
\]
3. Apply the power rule of exponents to both \(a^{-5}\) and \(b^6\):
\[
\left(a^{-5}b^6\right)^{-\frac{2}{3}} = a^{-5 \cdot -\frac{2}{3}} \cdot b^{6 \cdot -\frac{2}{3}}
\]
4. Simplify the exponents:
- For \(a\):
\[
-5 \cdot -\frac{2}{3} = \frac{10}{3}
\]
- For \(b\):
\[
6 \cdot -\frac{2}{3} = -\frac{12}{3} = -4
\]
5. Combine the results:
\[
a^{\frac{10}{3}} \cdot b^{-4} = \frac{a^{\frac{10}{3}}}{b^4}
\]

#### Final Answer:
\[
\boxed{\frac{a^{\frac{10}{3}}}{b^4}}
\]

---

Problem 5: Simplify \(\left(\frac{m^2n^{-3}}{m^{-1}n^2}\right)^{\frac{3}{2}}\)

#### Solution:
1. Simplify the fraction inside the parentheses:
\[
\frac{m^2n^{-3}}{m^{-1}n^2} = m^{2-(-1)} \cdot n^{-3-2} = m^{2+1} \cdot n^{-3-2} = m^3 \cdot n^{-5}
\]
2. Rewrite the expression with the simplified fraction:
\[
\left(m^3n^{-5}\right)^{\frac{3}{2}}
\]
3. Apply the power rule of exponents to both \(m^3\) and \(n^{-5}\):
\[
\left(m^3n^{-5}\right)^{\frac{3}{2}} = m^{3 \cdot \frac{3}{2}} \cdot n^{-5 \cdot \frac{3}{2}}
\]
4. Simplify the exponents:
- For \(m\):
\[
3 \cdot \frac{3}{2} = \frac{9}{2}
\]
- For \(n\):
\[
-5 \cdot \frac{3}{2} = -\frac{15}{2}
\]
5. Combine the results:
\[
m^{\frac{9}{2}} \cdot n^{-\frac{15}{2}} = \frac{m^{\frac{9}{2}}}{n^{\frac{15}{2}}}
\]

#### Final Answer:
\[
\boxed{\frac{m^{\frac{9}{2}}}{n^{\frac{15}{2}}}}
\]

---

Problem 6: Simplify \(\left(\frac{r^{-4}s^5}{r^3s^{-2}}\right)^{-\frac{1}{2}}\)

#### Solution:
1. Simplify the fraction inside the parentheses:
\[
\frac{r^{-4}s^5}{r^3s^{-2}} = r^{-4-3} \cdot s^{5-(-2)} = r^{-7} \cdot s^{5+2} = r^{-7} \cdot s^7
\]
2. Rewrite the expression with the simplified fraction:
\[
\left(r^{-7}s^7\right)^{-\frac{1}{2}}
\]
3. Apply the power rule of exponents to both \(r^{-7}\) and \(s^7\):
\[
\left(r^{-7}s^7\right)^{-\frac{1}{2}} = r^{-7 \cdot -\frac{1}{2}} \cdot s^{7 \cdot -\frac{1}{2}}
\]
4. Simplify the exponents:
- For \(r\):
\[
-7 \cdot -\frac{1}{2} = \frac{7}{2}
\]
- For \(s\):
\[
7 \cdot -\frac{1}{2} = -\frac{7}{2}
\]
5. Combine the results:
\[
r^{\frac{7}{2}} \cdot s^{-\frac{7}{2}} = \frac{r^{\frac{7}{2}}}{s^{\frac{7}{2}}}
\]

#### Final Answer:
\[
\boxed{\frac{r^{\frac{7}{2}}}{s^{\frac{7}{2}}}}
\]

---

Problem 7: Simplify \(\left(\frac{t^3u^{-2}}{t^{-1}u^4}\right)^{\frac{3}{4}}\)

#### Solution:
1. Simplify the fraction inside the parentheses:
\[
\frac{t^3u^{-2}}{t^{-1}u^4} = t^{3-(-1)} \cdot u^{-2-4} = t^{3+1} \cdot u^{-2-4} = t^4 \cdot u^{-6}
\]
2. Rewrite the expression with the simplified fraction:
\[
\left(t^4u^{-6}\right)^{\frac{3}{4}}
\]
3. Apply the power rule of exponents to both \(t^4\) and \(u^{-6}\):
\[
\left(t^4u^{-6}\right)^{\frac{3}{4}} = t^{4 \cdot \frac{3}{4}} \cdot u^{-6 \cdot \frac{3}{4}}
\]
4. Simplify the exponents:
- For \(t\):
\[
4 \cdot \frac{3}{4} = 3
\]
- For \(u\):
\[
-6 \cdot \frac{3}{4} = -\frac{18}{4} = -\frac{9}{2}
\]
5. Combine the results:
\[
t^3 \cdot u^{-\frac{9}{2}} = \frac{t^3}{u^{\frac{9}{2}}}
\]

#### Final Answer:
\[
\boxed{\frac{t^3}{u^{\frac{9}{2}}}}
\]

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Problem 8: Simplify \(\left(\frac{v^{-5}w^4}{v^2w^{-3}}\right)^{-\frac{2}{3}}\)

#### Solution:
1. Simplify the fraction inside the parentheses:
\[
\frac{v^{-5}w^4}{v^2w^{-3}} = v^{-5-2} \cdot w^{4-(-3)} = v^{-7} \cdot w^{4+3} = v^{-7} \cdot w^7
\]
2. Rewrite the expression with the simplified fraction:
\[
\left(v^{-7}w^7\right)^{-\frac{2}{3}}
\]
3. Apply the power rule of exponents to both \(v^{-7}\) and \(w^7\):
\[
\left(v^{-7}w^7\right)^{-\frac{2}{3}} = v^{-7 \cdot -\frac{2}{3}} \cdot w^{7 \cdot -\frac{2}{3}}
\]
4. Simplify the exponents:
- For \(v\):
\[
-7 \cdot -\frac{2}{3} = \frac{14}{3}
\]
- For \(w\):
\[
7 \cdot -\frac{2}{3} = -\frac{14}{3}
\]
5. Combine the results:
\[
v^{\frac{14}{3}} \cdot w^{-\frac{14}{3}} = \frac{v^{\frac{14}{3}}}{w^{\frac{14}{3}}}
\]

#### Final Answer:
\[
\boxed{\frac{v^{\frac{14}{3}}}{w^{\frac{14}{3}}}}
\]

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Problem 9: Simplify \(\left(\frac{x^{-2}y^3}{x^4y^{-1}}\right)^{-\frac{1}{2}}\)

This is the same as Problem 3. The solution is already provided above.

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

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Problem 10: Simplify \(\left(\frac{z^{-3}w^2}{z^2w^{-4}}\right)^{-\frac{2}{3}}\)

This is the same as Problem 4. The solution is already provided above.

#### Final Answer:
\[
\boxed{\frac{z^{\frac{10}{3}}}{w^4}}
\]

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Final Answers Summary:

1. \(\boxed{n^6}\)
2. \(\boxed{243p^{10}}\)
3. \(\boxed{\frac{x^3}{y^2}}\)
4. \(\boxed{\frac{a^{\frac{10}{3}}}{b^4}}\)
5. \(\boxed{\frac{m^{\frac{9}{2}}}{n^{\frac{15}{2}}}}\)
6. \(\boxed{\frac{r^{\frac{7}{2}}}{s^{\frac{7}{2}}}}\)
7. \(\boxed{\frac{t^3}{u^{\frac{9}{2}}}}\)
8. \(\boxed{\frac{v^{\frac{14}{3}}}{w^{\frac{14}{3}}}}\)
9. \(\boxed{\frac{x^3}{y^2}}\)
10. \(\boxed{\frac{z^{\frac{10}{3}}}{w^4}}\)