1. Simplify the following expression using positive exponents.
- $\frac{(p^{-\frac{3}{4}}q^3)^{\frac{1}{3}}}{(p^{-2}q^4)(p^2q^4)^{\frac{1}{2}}}$
- First, simplify the numerator: $(p^{-\frac{3}{4}}q^3)^{\frac{1}{3}} = p^{-\frac{3}{4} \cdot \frac{1}{3}}q^{3 \cdot \frac{1}{3}} = p^{-\frac{1}{4}}q^1$
- Next, simplify the denominator: $(p^{-2}q^4)(p^2q^4)^{\frac{1}{2}} = (p^{-2}q^4)(p^{2 \cdot \frac{1}{2}}q^{4 \cdot \frac{1}{2}}) = (p^{-2}q^4)(p^1q^2) = p^{-2+1}q^{4+2} = p^{-1}q^6$
- Now, divide the numerator by the denominator: $\frac{p^{-\frac{1}{4}}q}{p^{-1}q^6} = p^{-\frac{1}{4} - (-1)}q^{1 - 6} = p^{-\frac{1}{4} + 1}q^{-5} = p^{\frac{3}{4}}q^{-5}$
- Convert to positive exponents: $\frac{p^{\frac{3}{4}}}{q^5}$
- The correct answer is: $\frac{p^{\frac{3}{4}}}{q^5}$
2. Simplify the expression below using positive exponents.
- $\frac{(p^{\frac{3}{4}}q^2)^{\frac{1}{3}}}{(p^{-\frac{2}{3}}q^{\frac{1}{2}})(pq)^{-\frac{1}{3}}}$
- First, simplify the numerator: $(p^{\frac{3}{4}}q^2)^{\frac{1}{3}} = p^{\frac{3}{4} \cdot \frac{1}{3}}q^{2 \cdot \frac{1}{3}} = p^{\frac{1}{4}}q^{\frac{2}{3}}$
- Next, simplify the denominator: $(p^{-\frac{2}{3}}q^{\frac{1}{2}})(pq)^{-\frac{1}{3}} = (p^{-\frac{2}{3}}q^{\frac{1}{2}})(p^{-\frac{1}{3}}q^{-\frac{1}{3}}) = p^{-\frac{2}{3} - \frac{1}{3}}q^{\frac{1}{2} - \frac{1}{3}} = p^{-1}q^{\frac{1}{6}}$
- Now, divide the numerator by the denominator: $\frac{p^{\frac{1}{4}}q^{\frac{2}{3}}}{p^{-1}q^{\frac{1}{6}}} = p^{\frac{1}{4} - (-1)}q^{\frac{2}{3} - \frac{1}{6}} = p^{\frac{1}{4} + 1}q^{\frac{4}{6} - \frac{1}{6}} = p^{\frac{5}{4}}q^{\frac{1}{2}}$
- The correct answer is: $p^{\frac{5}{4}}q^{\frac{1}{2}}$
3. Using positive exponents, simplify the expression.
- $\left(\frac{m^{\frac{1}{2}}n^{\frac{2}{3}}}{m^{\frac{2}{5}}n^{\frac{3}{4}}}\right)^{-6}$
- First, simplify inside the parentheses: $\frac{m^{\frac{1}{2}}n^{\frac{2}{3}}}{m^{\frac{2}{5}}n^{\frac{3}{4}}} = m^{\frac{1}{2} - \frac{2}{5}}n^{\frac{2}{3} - \frac{3}{4}} = m^{\frac{5}{10} - \frac{4}{10}}n^{\frac{8}{12} - \frac{9}{12}} = m^{\frac{1}{10}}n^{-\frac{1}{12}}$
- Now, apply the exponent of -6: $(m^{\frac{1}{10}}n^{-\frac{1}{12}})^{-6} = m^{\frac{1}{10} \cdot (-6)}n^{-\frac{1}{12} \cdot (-6)} = m^{-\frac{6}{10}}n^{\frac{6}{12}} = m^{-\frac{3}{5}}n^{\frac{1}{2}}$
- Convert to positive exponents: $\frac{n^{\frac{1}{2}}}{m^{\frac{3}{5}}}$
- The correct answer is: $\frac{n^{\frac{1}{2}}}{m^{\frac{3}{5}}}$
Parent Tip: Review the logic above to help your child master the concept of rational exponent worksheet.
