Radicals and Rational Exponents Worksheets - Math Monks - Free Printable

Let's solve each problem by converting rational exponents to radical notation.

---

🔁 Rule for Converting Rational Exponents to Radicals:

For any expression $ a^{\frac{m}{n}} $, the radical form is:

$$
a^{\frac{m}{n}} = \sqrt[n]{a^m} \quad \text{or} \quad (\sqrt[n]{a})^m
$$

If the exponent is negative:
$$
a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}
$$

Also, if the base includes parentheses (like $(3x)^{\frac{3}{5}}$), the entire expression inside the parentheses is under the radical.

---

Now let's go through each one:

---

1. $-2^{\frac{3}{4}}$

⚠️ Important: This means negative of $2^{\frac{3}{4}}$, not $(-2)^{\frac{3}{4}}$, because the negative sign is outside.

So:
$$
-2^{\frac{3}{4}} = -\sqrt[4]{2^3} = -\sqrt[4]{8}
$$

✔ Answer: $-\sqrt[4]{8}$

---

2. $(3x)^{\frac{3}{5}}$

The entire base $3x$ is raised to the power $\frac{3}{5}$, so:
$$
(3x)^{\frac{3}{5}} = \sqrt[5]{(3x)^3}
$$

✔ Answer: $\sqrt[5]{(3x)^3}$

---

3. $x^{\frac{3}{6}}$

Simplify the exponent: $\frac{3}{6} = \frac{1}{2}$

So:
$$
x^{\frac{1}{2}} = \sqrt{x}
$$

✔ Answer: $\sqrt{x}$

---

4. $100^{\frac{1}{2}}$

$$
100^{\frac{1}{2}} = \sqrt{100}
$$

✔ Answer: $\sqrt{100}$

---

5. $a^{\frac{4}{3}}$

$$
a^{\frac{4}{3}} = \sqrt[3]{a^4}
$$

✔ Answer: $\sqrt[3]{a^4}$

---

6. $8^{-\frac{3}{4}}$

Negative exponent → reciprocal:
$$
8^{-\frac{3}{4}} = \frac{1}{8^{\frac{3}{4}}} = \frac{1}{\sqrt[4]{8^3}}
$$

✔ Answer: $\frac{1}{\sqrt[4]{512}}$ (since $8^3 = 512$)

Alternatively: $\frac{1}{\sqrt[4]{8^3}}$

---

7. $x^{\frac{5}{2}}$

$$
x^{\frac{5}{2}} = \sqrt{x^5} \quad \text{or} \quad (\sqrt{x})^5
$$

✔ Answer: $\sqrt{x^5}$

---

8. $(6x)^{\frac{5}{4}}$

$$
(6x)^{\frac{5}{4}} = \sqrt[4]{(6x)^5}
$$

✔ Answer: $\sqrt[4]{(6x)^5}$

---

9. $(10n)^{\frac{3}{2}}$

$$
(10n)^{\frac{3}{2}} = \sqrt{(10n)^3}
$$

✔ Answer: $\sqrt{(10n)^3}$

---

10. $m^{-\frac{1}{2}}$

$$
m^{-\frac{1}{2}} = \frac{1}{m^{\frac{1}{2}}} = \frac{1}{\sqrt{m}}
$$

✔ Answer: $\frac{1}{\sqrt{m}}$

---

11. $(10a)^{\frac{4}{5}}$

$$
(10a)^{\frac{4}{5}} = \sqrt[5]{(10a)^4}
$$

✔ Answer: $\sqrt[5]{(10a)^4}$

---

12. $x^{\frac{5}{2}}$

Same as #7:
$$
x^{\frac{5}{2}} = \sqrt{x^5}
$$

✔ Answer: $\sqrt{x^5}$

---

13. $(253)^{\frac{7}{5}}$

$$
(253)^{\frac{7}{5}} = \sqrt[5]{253^7}
$$

✔ Answer: $\sqrt[5]{253^7}$

---

14. $\left(\frac{45}{8}\right)^{\frac{2}{9}}$

$$
\left(\frac{45}{8}\right)^{\frac{2}{9}} = \sqrt[9]{\left(\frac{45}{8}\right)^2}
$$

✔ Answer: $\sqrt[9]{\left(\frac{45}{8}\right)^2}$

---

15. $\left(\frac{17}{231}\right)^{-\frac{5}{6}}$

Negative exponent → reciprocal:
$$
= \frac{1}{\left(\frac{17}{231}\right)^{\frac{5}{6}}} = \frac{1}{\sqrt[6]{\left(\frac{17}{231}\right)^5}}
$$

Alternatively, write as:
$$
= \left(\frac{231}{17}\right)^{\frac{5}{6}} = \sqrt[6]{\left(\frac{231}{17}\right)^5}
$$

✔ Answer: $\sqrt[6]{\left(\frac{231}{17}\right)^5}$

---

16. $(100)^{\frac{5}{6}}$

$$
100^{\frac{5}{6}} = \sqrt[6]{100^5}
$$

✔ Answer: $\sqrt[6]{100^5}$

---

17. $(21)^{\frac{1}{8}}$

$$
= \sqrt[8]{21}
$$

✔ Answer: $\sqrt[8]{21}$

---

18. $\left(\frac{15}{21}\right)^{\frac{2}{5}}$

Simplify fraction first: $\frac{15}{21} = \frac{5}{7}$

So:
$$
\left(\frac{5}{7}\right)^{\frac{2}{5}} = \sqrt[5]{\left(\frac{5}{7}\right)^2}
$$

✔ Answer: $\sqrt[5]{\left(\frac{5}{7}\right)^2}$

---

✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $-\sqrt[4]{8}$ |
| 2 | $\sqrt[5]{(3x)^3}$ |
| 3 | $\sqrt{x}$ |
| 4 | $\sqrt{100}$ |
| 5 | $\sqrt[3]{a^4}$ |
| 6 | $\frac{1}{\sqrt[4]{512}}$ or $\frac{1}{\sqrt[4]{8^3}}$ |
| 7 | $\sqrt{x^5}$ |
| 8 | $\sqrt[4]{(6x)^5}$ |
| 9 | $\sqrt{(10n)^3}$ |
| 10 | $\frac{1}{\sqrt{m}}$ |
| 11 | $\sqrt[5]{(10a)^4}$ |
| 12 | $\sqrt{x^5}$ |
| 13 | $\sqrt[5]{253^7}$ |
| 14 | $\sqrt[9]{\left(\frac{45}{8}\right)^2}$ |
| 15 | $\sqrt[6]{\left(\frac{231}{17}\right)^5}$ |
| 16 | $\sqrt[6]{100^5}$ |
| 17 | $\sqrt[8]{21}$ |
| 18 | $\sqrt[5]{\left(\frac{5}{7}\right)^2}$ |

---

Let me know if you'd like these simplified further or written in a specific format!