Simplifying expressions with rational exponents worksheet with multiple-choice questions.

Problem 1: Simplify the following expression using positive exponents.

\[
\frac{\left(p^{-\frac{3}{4}} q^{3}\right)^{\frac{1}{3}}}{(p^{-2} q^{4}) \left(p^{2} q^{4}\right)^{\frac{1}{2}}}
\]

#### Step-by-Step Solution:
1. Simplify the numerator:
\[
\left(p^{-\frac{3}{4}} q^{3}\right)^{\frac{1}{3}}
\]
Using the power of a product rule \((a^m b^n)^k = a^{mk} b^{nk}\):
\[
\left(p^{-\frac{3}{4}}\right)^{\frac{1}{3}} \left(q^{3}\right)^{\frac{1}{3}} = p^{-\frac{3}{4} \cdot \frac{1}{3}} q^{3 \cdot \frac{1}{3}} = p^{-\frac{1}{4}} q^{1}
\]

2. Simplify the denominator:
\[
(p^{-2} q^{4}) \left(p^{2} q^{4}\right)^{\frac{1}{2}}
\]
First, simplify \(\left(p^{2} q^{4}\right)^{\frac{1}{2}}\):
\[
\left(p^{2}\right)^{\frac{1}{2}} \left(q^{4}\right)^{\frac{1}{2}} = p^{2 \cdot \frac{1}{2}} q^{4 \cdot \frac{1}{2}} = p^{1} q^{2}
\]
Now, multiply \(p^{-2} q^{4}\) by \(p^{1} q^{2}\):
\[
(p^{-2} q^{4}) (p^{1} q^{2}) = p^{-2 + 1} q^{4 + 2} = p^{-1} q^{6}
\]

3. Combine the numerator and denominator:
\[
\frac{p^{-\frac{1}{4}} q^{1}}{p^{-1} q^{6}}
\]
Using the quotient rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\):
\[
p^{-\frac{1}{4} - (-1)} q^{1 - 6} = p^{-\frac{1}{4} + 1} q^{-5} = p^{\frac{3}{4}} q^{-5}
\]

4. Express with positive exponents:
\[
p^{\frac{3}{4}} q^{-5} = \frac{p^{\frac{3}{4}}}{q^{5}}
\]

#### Final Answer:
\[
\boxed{\frac{p^{\frac{3}{4}}}{q^{5}}}
\]

---

Problem 2: Simplify the expression below using positive exponents.

\[
\frac{\left(p^{\frac{3}{4}} q^{2}\right)^{\frac{1}{3}}}{\left(p^{-\frac{2}{3}} q^{\frac{1}{2}}\right) (pq)^{-\frac{1}{3}}}
\]

#### Step-by-Step Solution:
1. Simplify the numerator:
\[
\left(p^{\frac{3}{4}} q^{2}\right)^{\frac{1}{3}}
\]
Using the power of a product rule:
\[
\left(p^{\frac{3}{4}}\right)^{\frac{1}{3}} \left(q^{2}\right)^{\frac{1}{3}} = p^{\frac{3}{4} \cdot \frac{1}{3}} q^{2 \cdot \frac{1}{3}} = p^{\frac{1}{4}} q^{\frac{2}{3}}
\]

2. Simplify the denominator:
\[
\left(p^{-\frac{2}{3}} q^{\frac{1}{2}}\right) (pq)^{-\frac{1}{3}}
\]
First, simplify \((pq)^{-\frac{1}{3}}\):
\[
(pq)^{-\frac{1}{3}} = p^{-\frac{1}{3}} q^{-\frac{1}{3}}
\]
Now, multiply \(\left(p^{-\frac{2}{3}} q^{\frac{1}{2}}\right)\) by \(p^{-\frac{1}{3}} q^{-\frac{1}{3}}\):
\[
\left(p^{-\frac{2}{3}} q^{\frac{1}{2}}\right) \left(p^{-\frac{1}{3}} q^{-\frac{1}{3}}\right) = p^{-\frac{2}{3} - \frac{1}{3}} q^{\frac{1}{2} - \frac{1}{3}} = p^{-1} q^{\frac{1}{6}}
\]

3. Combine the numerator and denominator:
\[
\frac{p^{\frac{1}{4}} q^{\frac{2}{3}}}{p^{-1} q^{\frac{1}{6}}}
\]
Using the quotient rule for exponents:
\[
p^{\frac{1}{4} - (-1)} q^{\frac{2}{3} - \frac{1}{6}} = p^{\frac{1}{4} + 1} q^{\frac{2}{3} - \frac{1}{6}}
\]
Simplify the exponents:
\[
p^{\frac{1}{4} + \frac{4}{4}} = p^{\frac{5}{4}}
\]
\[
q^{\frac{2}{3} - \frac{1}{6}} = q^{\frac{4}{6} - \frac{1}{6}} = q^{\frac{3}{6}} = q^{\frac{1}{2}}
\]

4. Final expression:
\[
p^{\frac{5}{4}} q^{\frac{1}{2}}
\]

#### Final Answer:
\[
\boxed{p^{\frac{5}{4}} q^{\frac{1}{2}}}
\]

---

Problem 3: Using positive exponents, simplify the expression.

\[
\left(\frac{m^{\frac{1}{2}} n^{\frac{2}{3}}}{m^{\frac{2}{5}} n^{\frac{3}{4}}}\right)^{-6}
\]

#### Step-by-Step Solution:
1. Simplify the fraction inside the parentheses:
\[
\frac{m^{\frac{1}{2}} n^{\frac{2}{3}}}{m^{\frac{2}{5}} n^{\frac{3}{4}}}
\]
Using the quotient rule for exponents:
\[
m^{\frac{1}{2} - \frac{2}{5}} n^{\frac{2}{3} - \frac{3}{4}}
\]
Simplify the exponents for \(m\):
\[
\frac{1}{2} - \frac{2}{5} = \frac{5}{10} - \frac{4}{10} = \frac{1}{10}
\]
Simplify the exponents for \(n\):
\[
\frac{2}{3} - \frac{3}{4} = \frac{8}{12} - \frac{9}{12} = -\frac{1}{12}
\]
So, the expression becomes:
\[
m^{\frac{1}{10}} n^{-\frac{1}{12}}
\]

2. Apply the exponent \(-6\) to the simplified expression:
\[
\left(m^{\frac{1}{10}} n^{-\frac{1}{12}}\right)^{-6}
\]
Using the power of a product rule:
\[
\left(m^{\frac{1}{10}}\right)^{-6} \left(n^{-\frac{1}{12}}\right)^{-6} = m^{\frac{1}{10} \cdot (-6)} n^{-\frac{1}{12} \cdot (-6)} = m^{-\frac{6}{10}} n^{\frac{6}{12}}
\]
Simplify the exponents:
\[
m^{-\frac{6}{10}} = m^{-\frac{3}{5}}, \quad n^{\frac{6}{12}} = n^{\frac{1}{2}}
\]

3. Express with positive exponents:
\[
m^{-\frac{3}{5}} n^{\frac{1}{2}} = \frac{n^{\frac{1}{2}}}{m^{\frac{3}{5}}}
\]

#### Final Answer:
\[
\boxed{\frac{n^{\frac{1}{2}}}{m^{\frac{3}{5}}}}
\]