Rational Exponents Worksheets - Math Monks - Free Printable

Rational Exponents Practice Worksheet Solution

#### Part 1: Write each expression in radical form

1. Expression: \((6a)^{\frac{4}{3}}\)
- Solution: The exponent \(\frac{4}{3}\) indicates a cube root (denominator) and then raising to the 4th power (numerator).
\[
(6a)^{\frac{4}{3}} = \sqrt[3]{(6a)^4}
\]

2. Expression: \((10p)^{\frac{3}{4}}\)
- Solution: The exponent \(\frac{3}{4}\) indicates a fourth root (denominator) and then raising to the 3rd power (numerator).
\[
(10p)^{\frac{3}{4}} = \sqrt[4]{(10p)^3}
\]

3. Expression: \(4^{\frac{1}{6}}\)
- Solution: The exponent \(\frac{1}{6}\) indicates a sixth root.
\[
4^{\frac{1}{6}} = \sqrt[6]{4}
\]

---

#### Part 2: Write each expression in exponential form

4. Expression: \(\frac{1}{(\sqrt[3]{n})^7}\)
- Solution: The cube root \(\sqrt[3]{n}\) can be written as \(n^{\frac{1}{3}}\). Raising it to the 7th power gives \((n^{\frac{1}{3}})^7 = n^{\frac{7}{3}}\). The reciprocal is \(n^{-\frac{7}{3}}\).
\[
\frac{1}{(\sqrt[3]{n})^7} = n^{-\frac{7}{3}}
\]

5. Expression: \(\sqrt{6y}\)
- Solution: The square root can be written as an exponent of \(\frac{1}{2}\).
\[
\sqrt{6y} = (6y)^{\frac{1}{2}}
\]

6. Expression: \(\frac{1}{(\sqrt{6x})^3}\)
- Solution: The square root \(\sqrt{6x}\) can be written as \((6x)^{\frac{1}{2}}\). Raising it to the 3rd power gives \((6x)^{\frac{3}{2}}\). The reciprocal is \((6x)^{-\frac{3}{2}}\).
\[
\frac{1}{(\sqrt{6x})^3} = (6x)^{-\frac{3}{2}}
\]

---

#### Part 3: Simplify. Your answer should contain only positive exponents

7. Expression: \(\frac{x^2 y^0}{3x^4}\)
- Solution: Recall that \(y^0 = 1\). Simplify the \(x\)-terms using the rule \(x^a / x^b = x^{a-b}\):
\[
\frac{x^2 y^0}{3x^4} = \frac{x^2 \cdot 1}{3x^4} = \frac{x^2}{3x^4} = \frac{1}{3x^{4-2}} = \frac{1}{3x^2}
\]

8. Expression: \((x^0 y)^{\frac{3}{2}} x^0\)
- Solution: Recall that \(x^0 = 1\). Thus, the expression simplifies to:
\[
(x^0 y)^{\frac{3}{2}} x^0 = (1 \cdot y)^{\frac{3}{2}} \cdot 1 = y^{\frac{3}{2}}
\]

9. Expression: \(4x^{\frac{2}{3}} x^{-1}\)
- Solution: Combine the \(x\)-terms using the rule \(x^a \cdot x^b = x^{a+b}\):
\[
4x^{\frac{2}{3}} x^{-1} = 4x^{\frac{2}{3} + (-1)} = 4x^{\frac{2}{3} - \frac{3}{3}} = 4x^{-\frac{1}{3}}
\]
Convert to positive exponent:
\[
4x^{-\frac{1}{3}} = \frac{4}{x^{\frac{1}{3}}}
\]

10. Expression: \(\frac{4x^2}{2x^{\frac{1}{2}}}\)
- Solution: Simplify the coefficients and the \(x\)-terms separately:
\[
\frac{4x^2}{2x^{\frac{1}{2}}} = \frac{4}{2} \cdot \frac{x^2}{x^{\frac{1}{2}}} = 2 \cdot x^{2 - \frac{1}{2}} = 2 \cdot x^{\frac{4}{2} - \frac{1}{2}} = 2x^{\frac{3}{2}}
\]

11. Expression: \((s \cdot s^{-2} p^{\frac{5}{3}})^2\)
- Solution: Simplify inside the parentheses first:
\[
s \cdot s^{-2} = s^{1 + (-2)} = s^{-1}
\]
So the expression becomes:
\[
(s^{-1} p^{\frac{5}{3}})^2
\]
Apply the power of a product rule \((ab)^n = a^n b^n\):
\[
(s^{-1} p^{\frac{5}{3}})^2 = (s^{-1})^2 \cdot (p^{\frac{5}{3}})^2 = s^{-2} \cdot p^{\frac{10}{3}}
\]
Convert to positive exponent for \(s\):
\[
s^{-2} \cdot p^{\frac{10}{3}} = \frac{p^{\frac{10}{3}}}{s^2}
\]

12. Expression: \(\left(\frac{a^{\frac{1}{2}} b^{-2}}{ba^{-\frac{1}{4}}}\right)^4\)
- Solution: Simplify the fraction inside the parentheses first. Combine the \(a\)-terms and \(b\)-terms separately:
\[
\frac{a^{\frac{1}{2}} b^{-2}}{ba^{-\frac{1}{4}}} = \frac{a^{\frac{1}{2}}}{a^{-\frac{1}{4}}} \cdot \frac{b^{-2}}{b^1} = a^{\frac{1}{2} - (-\frac{1}{4})} \cdot b^{-2 - 1}
\]
Simplify the exponents:
\[
a^{\frac{1}{2} + \frac{1}{4}} \cdot b^{-3} = a^{\frac{2}{4} + \frac{1}{4}} \cdot b^{-3} = a^{\frac{3}{4}} \cdot b^{-3}
\]
Now raise the entire expression to the 4th power:
\[
\left(a^{\frac{3}{4}} b^{-3}\right)^4 = (a^{\frac{3}{4}})^4 \cdot (b^{-3})^4 = a^{\frac{3}{4} \cdot 4} \cdot b^{-3 \cdot 4} = a^3 \cdot b^{-12}
\]
Convert to positive exponent for \(b\):
\[
a^3 \cdot b^{-12} = \frac{a^3}{b^{12}}
\]

---

Final Answers:

1. \(\boxed{\sqrt[3]{(6a)^4}}\)
2. \(\boxed{\sqrt[4]{(10p)^3}}\)
3. \(\boxed{\sqrt[6]{4}}\)
4. \(\boxed{n^{-\frac{7}{3}}}\)
5. \(\boxed{(6y)^{\frac{1}{2}}}\)
6. \(\boxed{(6x)^{-\frac{3}{2}}}\)
7. \(\boxed{\frac{1}{3x^2}}\)
8. \(\boxed{y^{\frac{3}{2}}}\)
9. \(\boxed{\frac{4}{x^{\frac{1}{3}}}}\)
10. \(\boxed{2x^{\frac{3}{2}}}\)
11. \(\boxed{\frac{p^{\frac{10}{3}}}{s^2}}\)
12. \(\boxed{\frac{a^3}{b^{12}}}\)

Final Answer Boxed:
\[
\boxed{
\begin{aligned}
&1. \sqrt[3]{(6a)^4} \\
&2. \sqrt[4]{(10p)^3} \\
&3. \sqrt[6]{4} \\
&4. n^{-\frac{7}{3}} \\
&5. (6y)^{\frac{1}{2}} \\
&6. (6x)^{-\frac{3}{2}} \\
&7. \frac{1}{3x^2} \\
&8. y^{\frac{3}{2}} \\
&9. \frac{4}{x^{\frac{1}{3}}} \\
&10. 2x^{\frac{3}{2}} \\
&11. \frac{p^{\frac{10}{3}}}{s^2} \\
&12. \frac{a^3}{b^{12}}
\end{aligned}
}
\]