Math worksheet on radicals and rational exponents, featuring exercises to convert between forms and simplify expressions.

Let's solve each part of the worksheet step by step. We'll use the rules of radicals and rational exponents:

🔷 Key Rules:

1. $ a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m $
2. $ \sqrt[n]{a} = a^{\frac{1}{n}} $
3. $ \left(\sqrt[n]{a}\right)^m = a^{\frac{m}{n}} $
4. $ \frac{1}{\sqrt[n]{a}} = a^{-\frac{1}{n}} $

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## ✔ Problem 1: Write each expression in radical form.

a) $ 2^{\frac{1}{6}} $

This means the 6th root of 2.

$$
2^{\frac{1}{6}} = \sqrt[6]{2}
$$

b) $ 27^{-\frac{2}{3}} $

Negative exponent means reciprocal, so:

$$
27^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}} = \frac{1}{\sqrt[3]{27^2}}
$$

But we can write it directly as:

$$
= \frac{1}{(\sqrt[3]{27})^2}
$$

Since $ \sqrt[3]{27} = 3 $, but for radical form, just leave it:

$$
= \frac{1}{\sqrt[3]{27^2}} \quad \text{or} \quad \frac{1}{(\sqrt[3]{27})^2}
$$

But the question asks for radical form, so:

$$
27^{-\frac{2}{3}} = \frac{1}{\sqrt[3]{27^2}} = \frac{1}{\sqrt[3]{729}}
$$

Alternatively, more standard:

$$
= \frac{1}{(\sqrt[3]{27})^2} = \frac{1}{(3)^2} = \frac{1}{9} \quad \text{(but that's simplified)}
$$

But since it says "radical form", keep it with radicals:

✔ Final answer:
$$
\boxed{\frac{1}{\sqrt[3]{27^2}}}
$$

Or better:
$$
\boxed{\frac{1}{(\sqrt[3]{27})^2}}
$$

But typically we write:
$$
\boxed{\frac{1}{\sqrt[3]{729}}}
$$

However, most acceptable is:
$$
\boxed{ \frac{1}{\left( \sqrt[3]{27} \right)^2 } }
$$

c) $ 1000^{\frac{6}{1}} $

Wait: $ 1000^6 $ — this is already an integer exponent.

But it's written as $ 1000^6 $, which is not a fractional exponent.

Wait — check:
It says $ 1000^6 $? But in the image, it’s likely meant to be $ 1000^{\frac{6}{1}} $? No — actually, it's probably $ 1000^{\frac{1}{6}} $? Wait, let's recheck.

Looking at your image:
It says: c) $ 1000^6 $ — but that's not a rational exponent. It's just $ 1000^6 $. That doesn't need radical form.

Wait — perhaps it's a typo? Or maybe it's $ 1000^{1/6} $? Let me assume it's $ 1000^{\frac{1}{6}} $ because otherwise it doesn’t make sense.

But based on what you wrote: c) $ 1000^6 $ — that's just $ 1000^6 $, which is already exponential form.

But the instruction says: “Write each expression in radical form.”

So unless it's $ 1000^{1/6} $, this is strange.

Wait — looking again: it's written as $ 1000^6 $ — but that's not a rational exponent (it's an integer). So perhaps it's a mistake?

Wait — maybe it's $ 1000^{\frac{1}{6}} $? That would make sense.

Let me double-check: In many such worksheets, they write $ 1000^{1/6} $, meaning sixth root of 1000.

Assuming c) is $ 1000^{\frac{1}{6}} $, then:

$$
1000^{\frac{1}{6}} = \sqrt[6]{1000}
$$

✔ Answer: $ \boxed{\sqrt[6]{1000}} $

But if it's really $ 1000^6 $, then it's not a radical form problem — it's already simple. But since it's under "radical form", I believe it's likely $ 1000^{1/6} $. So we'll go with that.

d) $ 6^{\frac{2}{3}} $

$$
= \sqrt[3]{6^2} = \sqrt[3]{36}
$$

✔ Answer: $ \boxed{\sqrt[3]{36}} $

e) $ 8^{\frac{1}{5}} $

$$
= \sqrt[5]{8}
$$

✔ Answer: $ \boxed{\sqrt[5]{8}} $

f) $ 3x^{\frac{3}{4}} $

Note: This is 3 times $ x^{3/4} $, not $ (3x)^{3/4} $

So only the $ x $ has the exponent.

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

So:

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

✔ Answer: $ \boxed{3\sqrt[4]{x^3}} $

---

## ✔ Problem 2: Write each expression with a rational exponent.

a) $ \sqrt[4]{2} $

$$
= 2^{\frac{1}{4}}
$$

✔ Answer: $ \boxed{2^{\frac{1}{4}}} $

b) $ \sqrt[3]{2} $

$$
= 2^{\frac{1}{3}}
$$

✔ Answer: $ \boxed{2^{\frac{1}{3}}} $

c) $ \sqrt[6]{10} $

$$
= 10^{\frac{1}{6}}
$$

✔ Answer: $ \boxed{10^{\frac{1}{6}}} $

d) $ (\sqrt{2})^5 $

First, $ \sqrt{2} = 2^{\frac{1}{2}} $, so:

$$
(2^{\frac{1}{2}})^5 = 2^{\frac{5}{2}}
$$

✔ Answer: $ \boxed{2^{\frac{5}{2}}} $

e) $ (\sqrt{5})^5 $

Similarly: $ \sqrt{5} = 5^{\frac{1}{2}} $, so:

$$
(5^{\frac{1}{2}})^5 = 5^{\frac{5}{2}}
$$

✔ Answer: $ \boxed{5^{\frac{5}{2}}} $

f) $ (\sqrt{2})^5 $

Same as (d): $ 2^{\frac{5}{2}} $

✔ Answer: $ \boxed{2^{\frac{5}{2}}} $

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## ✔ Problem 3: Write each expression in exponential form.

a) $ (\sqrt{36})^4 $

First, $ \sqrt{36} = 36^{\frac{1}{2}} $, so:

$$
(36^{\frac{1}{2}})^4 = 36^{\frac{4}{2}} = 36^2
$$

But they want exponential form, so:

$$
= (36)^2
$$

But also, $ 36 = 6^2 $, so $ 36^2 = (6^2)^2 = 6^4 $, but not necessary.

We can write it as:

$$
(\sqrt{36})^4 = (36^{\frac{1}{2}})^4 = 36^{\frac{4}{2}} = 36^2
$$

✔ Answer: $ \boxed{36^2} $

Or even better: $ \boxed{(36)^2} $

But if they want rational exponent, but it says "exponential form" — so any exponent form is fine.

Alternatively, $ 36 = 6^2 $, so $ 36^2 = (6^2)^2 = 6^4 $, but simplest is $ 36^2 $.

But let's see if they want it in terms of base 6.

But no, $ 36^2 $ is fine.

✔ Final: $ \boxed{36^2} $

b) $ \sqrt{6p} $

This is $ (6p)^{\frac{1}{2}} $

✔ Answer: $ \boxed{(6p)^{\frac{1}{2}}} $

c) $ \frac{1}{(\sqrt{3x})^5} $

First, $ \sqrt{3x} = (3x)^{\frac{1}{2}} $, so:

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

Then:

$$
\frac{1}{(3x)^{\frac{5}{2}}} = (3x)^{-\frac{5}{2}}
$$

✔ Answer: $ \boxed{(3x)^{-\frac{5}{2}}} $

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## ✔ Problem 4: Simplify.

a) $ 100000^{\frac{1}{6}} $

First, factor $ 100000 $:

$$
100000 = 10^5
$$

So:

$$
(10^5)^{\frac{1}{6}} = 10^{\frac{5}{6}}
$$

Can we simplify further? Not really.

But is there a perfect 6th power?

Check: $ 10^5 = 100000 $

Try $ 10^{\frac{5}{6}} $ — this is $ \sqrt[6]{10^5} = \sqrt[6]{100000} $

But it doesn’t simplify nicely.

Alternatively, note:

Is $ 100000 = 10^5 $, and $ (10^5)^{1/6} = 10^{5/6} $

So simplified form is:

✔ $ \boxed{10^{\frac{5}{6}}} $

But let's see if it can be written differently.

Alternatively, $ 100000 = 10^5 = (10^5)^{1/6} = 10^{5/6} $

Yes.

So final: $ \boxed{10^{\frac{5}{6}}} $

b) $ (9x^4)^{\frac{1}{2}} $

Apply exponent to both parts:

$$
= 9^{\frac{1}{2}} \cdot (x^4)^{\frac{1}{2}} = \sqrt{9} \cdot x^{2} = 3x^2
$$

✔ Answer: $ \boxed{3x^2} $

c) $ (36n^{12})^{\frac{1}{6}} $

Break into parts:

$$
= 36^{\frac{1}{6}} \cdot (n^{12})^{\frac{1}{6}} = 36^{\frac{1}{6}} \cdot n^{2}
$$

Now simplify $ 36^{\frac{1}{6}} $:

$ 36 = 6^2 $, so:

$$
(6^2)^{\frac{1}{6}} = 6^{\frac{2}{6}} = 6^{\frac{1}{3}} = \sqrt[3]{6}
$$

So:

$$
= \sqrt[3]{6} \cdot n^2
$$

✔ Answer: $ \boxed{n^2 \sqrt[3]{6}} $

Or $ \boxed{n^2 \cdot 6^{1/3}} $

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## ✔ Final Answers Summary

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Problem 1: Radical Form

a) $ 2^{\frac{1}{6}} = \boxed{\sqrt[6]{2}} $
b) $ 27^{-\frac{2}{3}} = \boxed{\frac{1}{\sqrt[3]{27^2}}} $ or $ \boxed{\frac{1}{(\sqrt[3]{27})^2}} $
c) $ 1000^{\frac{1}{6}} = \boxed{\sqrt[6]{1000}} $ *(assuming typo, should be 1/6)*
d) $ 6^{\frac{2}{3}} = \boxed{\sqrt[3]{36}} $
e) $ 8^{\frac{1}{5}} = \boxed{\sqrt[5]{8}} $
f) $ 3x^{\frac{3}{4}} = \boxed{3\sqrt[4]{x^3}} $

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Problem 2: Rational Exponent Form

a) $ \sqrt[4]{2} = \boxed{2^{\frac{1}{4}}} $
b) $ \sqrt[3]{2} = \boxed{2^{\frac{1}{3}}} $
c) $ \sqrt[6]{10} = \boxed{10^{\frac{1}{6}}} $
d) $ (\sqrt{2})^5 = \boxed{2^{\frac{5}{2}}} $
e) $ (\sqrt{5})^5 = \boxed{5^{\frac{5}{2}}} $
f) $ (\sqrt{2})^5 = \boxed{2^{\frac{5}{2}}} $

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Problem 3: Exponential Form

a) $ (\sqrt{36})^4 = \boxed{36^2} $
b) $ \sqrt{6p} = \boxed{(6p)^{\frac{1}{2}}} $
c) $ \frac{1}{(\sqrt{3x})^5} = \boxed{(3x)^{-\frac{5}{2}}} $

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Problem 4: Simplify

a) $ 100000^{\frac{1}{6}} = \boxed{10^{\frac{5}{6}}} $
b) $ (9x^4)^{\frac{1}{2}} = \boxed{3x^2} $
c) $ (36n^{12})^{\frac{1}{6}} = \boxed{n^2 \cdot 6^{1/3}} $ or $ \boxed{n^2 \sqrt[3]{6}} $

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