Simplify the Nested Square Root Expression: √(√(5x⁴))

Q: Why can't I just cancel out the square root signs?

Square roots don't simply "cancel" when nested! Each $ \sqrt{} $ means "raise to the power of 1/2", so nested square roots multiply exponents: 1/2 × 1/2 = 1/4, giving you a fourth root.

Q: How do I know when to simplify the inner radical first?

Always simplify the innermost expression first! This makes the problem easier. For $ \sqrt{5x^4} $, since $ x^4 = (x^2)^2 $, we can take $ x^2 $ out of the square root.

Q: What's the difference between √[4]{5} and √{5}?

The small number (4) indicates a fourth root! $ \sqrt[4]{5} $ asks "what number times itself 4 times equals 5?", while $ \sqrt{5} $ asks "what number times itself 2 times equals 5?"

Q: Can I use fractional exponents instead?

Absolutely! $ \sqrt{\sqrt{5x^4}} $ becomes $ (5x^4)^{1/2 \cdot 1/2} = (5x^4)^{1/4} = 5^{1/4} \cdot x $. This gives the same answer: $ \sqrt[4]{5} \cdot x $.

Q: How do I check my answer is correct?

Work backwards! If your answer is $ \sqrt[4]{5} \cdot x $, then squaring it twice should give you $ 5x^4 $. Try: $ (\sqrt[4]{5} \cdot x)^2 = \sqrt{5} \cdot x^2 $, then $ (\sqrt{5} \cdot x^2)^2 = 5x^4 $ ✓

Q: What if x could be negative?

For this problem, we assume x ≥ 0 to keep all expressions real and positive. When x can be negative, you'd need absolute value signs: $ \sqrt[4]{5} \cdot |x| $.

Nested Radicals with Power Simplification

Complete the following exercise:

$\sqrt[]{\sqrt{5x^4}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's solve this problem together.

00:10 A regular square root is of order two.

00:15 If we have A to the power of B inside a root of order C,

00:20 It equals A to the power of B times C.

00:25 Let's use this formula and multiply the powers in our exercise.

00:37 With a root of A times B,

00:40 We can split it into the root of A times the root of B.

00:49 Now we'll apply this rule, and separate the roots in our problem.

00:58 If A to the power of B is in a root of order C,

01:02 It becomes A to the power of B divided by C.

01:06 We'll use this to divide the exponents in our task.

01:12 And that's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt[]{\sqrt{5x^4}}=$

Step-by-step solution

To solve the expression $\sqrt[]{\sqrt{5x^4}}$ , let's go step-by-step:

Step 1: Simplify the inner expression $\sqrt{5x^4}$ . Using the rule for square roots, we can rewrite $\sqrt{5x^4}$ as $(5x^4)^{1/2}$ . This expression can be further simplified to $5^{1/2} \cdot (x^4)^{1/2} = \sqrt{5} \cdot x^2$ .
Step 2: Take the square root of the simplified expression. This means we apply another square root to $\sqrt{5} \cdot x^2$ , resulting in $(\sqrt{5} \cdot x^2)^{1/2} = (\sqrt{5})^{1/2} \cdot (x^2)^{1/2}$ .
Step 3: Simplify each component: $\sqrt[4]{5} \cdot x$ . We find that $(\sqrt{5})^{1/2}$ simplifies to $\sqrt[4]{5}$ and $(x^2)^{1/2}$ to $x$ .

Therefore, the simplified expression is $\sqrt[4]{5} \cdot x$ .

Final Answer

$\sqrt[4]{5}\cdot x$

Key Points to Remember

Essential concepts to master this topic

Rule: Combine nested square roots by multiplying their exponents: 1/2 × 1/2 = 1/4
Technique: Convert $\sqrt{5x^4}$ to $\sqrt{5} \cdot x^2$ before taking outer root
Check: Verify $(\sqrt[4]{5} \cdot x)^4 = 5x^4$ matches the original expression ✓

Common Mistakes

Avoid these frequent errors

Canceling square roots incorrectly
Don't assume $\sqrt{\sqrt{5x^4}} = \sqrt{5x}$ by canceling one square root! This ignores the fact that nested radicals require multiplying exponents (1/2 × 1/2 = 1/4), not simple cancellation. Always work from the inside out and apply exponent rules correctly.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[5]{\sqrt[3]{5}}=$

FAQ

Everything you need to know about this question

Why can't I just cancel out the square root signs?

Square roots don't simply "cancel" when nested! Each $\sqrt{}$ means "raise to the power of 1/2", so nested square roots multiply exponents: 1/2 × 1/2 = 1/4, giving you a fourth root.

How do I know when to simplify the inner radical first?

Always simplify the innermost expression first! This makes the problem easier. For $\sqrt{5x^4}$ , since $x^4 = (x^2)^2$ , we can take $x^2$ out of the square root.

What's the difference between √[4]{5} and √{5}?

The small number (4) indicates a fourth root! $\sqrt[4]{5}$ asks "what number times itself 4 times equals 5?", while $\sqrt{5}$ asks "what number times itself 2 times equals 5?"

Can I use fractional exponents instead?

Absolutely! $\sqrt{\sqrt{5x^4}}$ becomes $(5x^4)^{1/2 \cdot 1/2} = (5x^4)^{1/4} = 5^{1/4} \cdot x$ . This gives the same answer: $\sqrt[4]{5} \cdot x$ .

How do I check my answer is correct?

Work backwards! If your answer is $\sqrt[4]{5} \cdot x$ , then squaring it twice should give you $5x^4$ . Try: $(\sqrt[4]{5} \cdot x)^2 = \sqrt{5} \cdot x^2$ , then $(\sqrt{5} \cdot x^2)^2 = 5x^4$ ✓

What if x could be negative?

For this problem, we assume x ≥ 0 to keep all expressions real and positive. When x can be negative, you'd need absolute value signs: $\sqrt[4]{5} \cdot |x|$ .

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