Solve Nested Radicals: Fourth Root of Cube Root with 49x

Q: Why do I convert radicals to exponents?

Fractional exponents make it much easier to apply the power rule! Instead of dealing with confusing nested radical symbols, you can use the simple rule (a^m)^n = a^(mn).

Q: Can I simplify the expression further?

The expression $ \sqrt[12]{49x} $ is already in its simplest radical form. Since 49 = 7² and we need a 12th root, we can't factor out perfect 12th powers.

Q: What if the inner expression was different?

The same method works! For example, $ \sqrt[5]{\sqrt[2]{8x}} = (8x)^{1/2 \cdot 1/5} = (8x)^{1/10} = \sqrt[10]{8x} $. Always multiply the fractional exponents.

Q: How do I check my answer?

Work backwards! Start with $ \sqrt[12]{49x} = (49x)^{1/12} $, then apply the fourth root: $ ((49x)^{1/12})^4 = (49x)^{4/12} = (49x)^{1/3} = \sqrt[3]{49x} $ ✓

Nested Radicals with Fractional Exponents

Complete the following exercise:

$\sqrt[4]{\sqrt[3]{49\cdot x}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's solve this math problem together.

00:10 When we have a number, A, raised to the power, B, under a root of order, C.

00:16 The result is the number, A, under a root of order, B times C.

00:22 Now, let's use this formula in our exercise.

00:25 First, calculate the product of the orders, B and C.

00:30 And that's how we find the solution. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt[4]{\sqrt[3]{49\cdot x}}=$

Step-by-step solution

To simplify the given expression $\sqrt[4]{\sqrt[3]{49 \cdot x}}$ , we will use the property of roots as fractional exponents.

Step 1: Convert the cube root to a fractional exponent. We have $\sqrt[3]{49 \cdot x} = (49 \cdot x)^{1/3}$ .
Step 2: Apply the fourth root to the result, expressed as a fractional exponent. Thus, $\sqrt[4]{(49 \cdot x)^{1/3}} = ((49 \cdot x)^{1/3})^{1/4}$ .
Step 3: Combine the exponents by multiplying the fractions: $((49 \cdot x)^{1/3})^{1/4} = (49 \cdot x)^{(1/3) \times (1/4)} = (49 \cdot x)^{1/12}$ .
Step 4: Recognize that the result $(49 \cdot x)^{1/12}$ can be expressed as $\sqrt[12]{49 \cdot x}$ .

Therefore, the solution to the problem is $\sqrt[12]{49x}$ .

Final Answer

$\sqrt[12]{49x}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested radicals to fractional exponents for easier manipulation
Technique: Multiply exponents: $(a^{1/3})^{1/4} = a^{1/3 \cdot 1/4} = a^{1/12}$
Check: Convert final answer back to radical form: $(49x)^{1/12} = \sqrt[12]{49x}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add the fractions 1/3 + 1/4 = 7/12! This gives $(49x)^{7/12}$ which is completely wrong. When raising a power to a power, always multiply the exponents: $(a^m)^n = a^{m \cdot n}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

FAQ

Everything you need to know about this question

Why do I convert radicals to exponents?

Fractional exponents make it much easier to apply the power rule! Instead of dealing with confusing nested radical symbols, you can use the simple rule (a^m)^n = a^(mn).

How do I multiply fractions like 1/3 × 1/4?

Multiply the numerators together and denominators together: $\frac{1}{3} \times \frac{1}{4} = \frac{1 \times 1}{3 \times 4} = \frac{1}{12}$ . This gives you the final exponent!

Can I simplify the expression further?

The expression $\sqrt[12]{49x}$ is already in its simplest radical form. Since 49 = 7² and we need a 12th root, we can't factor out perfect 12th powers.

What if the inner expression was different?

The same method works! For example, $\sqrt[5]{\sqrt[2]{8x}} = (8x)^{1/2 \cdot 1/5} = (8x)^{1/10} = \sqrt[10]{8x}$ . Always multiply the fractional exponents.

How do I check my answer?

Work backwards! Start with $\sqrt[12]{49x} = (49x)^{1/12}$ , then apply the fourth root: $((49x)^{1/12})^4 = (49x)^{4/12} = (49x)^{1/3} = \sqrt[3]{49x}$ ✓

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