Simplify the Nested Square Root: √(√(3x²))

Q: Why can't I just simplify the inner square root first?

When dealing with nested radicals, it's easier to work with exponents! Converting $\sqrt{\sqrt{3x^2}}$ to $((3x^2)^{1/2})^{1/2}$ lets you use the power rule directly.

Q: What's the difference between $\sqrt[4]{3}\cdot\sqrt{x}$ and $x\sqrt[4]{3}$ ?

The key difference is in the exponent of x! In $\sqrt[4]{3}\cdot\sqrt{x}$, we have $x^{1/2}$, but in $x\sqrt[4]{3}$ we have $x^1$. From $(x^2)^{1/4} = x^{2 \cdot 1/4} = x^{1/2}$, the first form is correct.

Q: How do I know when to use fractional exponents versus radicals?

Fractional exponents are usually easier for complex operations like nested radicals. Use them when applying exponent rules, then convert back to radical form for the final answer if needed.

Q: Why does $(x^2)^{1/4}$ equal $x^{1/2}$ and not just $x$ ?

Remember the power rule: $(a^m)^n = a^{mn}$. So $(x^2)^{1/4} = x^{2 \times 1/4} = x^{1/2}$. We multiply the exponents, not add them!

Q: Can I check my answer by substituting a value?

Yes! Try $x = 4$: $\sqrt{\sqrt{3 \cdot 16}} = \sqrt{\sqrt{48}} ≈ 2.63$, and $\sqrt[4]{3}\cdot\sqrt{4} = 1.316 \times 2 ≈ 2.63$ ✓

Nested Radicals with Exponent Rules

Complete the following exercise:

$\sqrt{\sqrt{3x^2}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 A "regular" root is of the order 2

00:11 When we have a number (A) to the power of (B) in a root of order (C)

00:14 The result equals the number (A) to the power of their quotient (B times C)

00:18 We will apply this formula to our exercise

00:24 Calculate the order multiplication

00:31 When we have a root of a product (A times B)

00:36 We can write it as the product of the root of each term

00:40 We will apply this formula to our exercise, and proceed to break down the root

00:49 When we have a number (A) to the power of (B) in a root of order (C)

00:52 The result equals the number (A) to the power of their quotient (B divided by C)

00:56 We will apply this formula to our exercise

01:04 Calculate the power quotient

01:12 We'll apply this formula again in the opposite direction, converting from a power to a root

01:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt{\sqrt{3x^2}}=$

Step-by-step solution

To solve $\sqrt{\sqrt{3x^2}}$ , follow these steps:

Step 1: Express the problem using exponentiation. The expression $\sqrt{3x^2}$ can be written as $(3x^2)^{\frac{1}{2}}$ .
Step 2: Take the square root of the first expression. This can be expressed as $((3x^2)^{\frac{1}{2}})^{\frac{1}{2}}$ .
Step 3: Use the property $(a^m)^n = a^{m \cdot n}$ . Thus, $((3x^2)^{\frac{1}{2}})^{\frac{1}{2}} = (3x^2)^{\frac{1}{4}}$ .
Step 4: Simplify further using exponent rules: $(3x^2)^{\frac{1}{4}}$ becomes $(3^{\frac{1}{4}} \cdot (x^2)^{\frac{1}{4}})$ , which simplifies to $\sqrt[4]{3} \cdot x^{\frac{1}{2}}$ .
Step 5: Recognize this as $\sqrt[4]{3} \cdot \sqrt{x}$ , since $x^{\frac{1}{2}}$ is $\sqrt{x}$ .

Therefore, the simplified form of the given expression is $\sqrt[4]{3} \cdot \sqrt{x}$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Power Rule: Convert nested square roots to fractional exponents first
Technique: Apply $(a^m)^n = a^{mn}$ so $(3x^2)^{1/2}$
Check: Verify by converting back to radical form: $(3x^2)^{1/4} = \sqrt[4]{3x^2}$ ✓

Common Mistakes

Avoid these frequent errors

Simplifying inner radical before applying outer radical
Don't simplify $\sqrt{3x^2}$ to $x\sqrt{3}$ first = wrong path! This ignores the absolute value requirement and leads to incorrect final answers. Always convert to fractional exponents and use the power rule $(a^m)^n = a^{mn}$ directly.

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 can't I just simplify the inner square root first?

When dealing with nested radicals, it's easier to work with exponents! Converting $\sqrt{\sqrt{3x^2}}$ to $((3x^2)^{1/2})^{1/2}$ lets you use the power rule directly.

What's the difference between $\sqrt[4]{3}\cdot\sqrt{x}$ and $x\sqrt[4]{3}$ ?

The key difference is in the exponent of x! In $\sqrt[4]{3}\cdot\sqrt{x}$ , we have $x^{1/2}$ , but in $x\sqrt[4]{3}$ we have $x^1$ . From $(x^2)^{1/4} = x^{2 \cdot 1/4} = x^{1/2}$ , the first form is correct.

How do I know when to use fractional exponents versus radicals?

Fractional exponents are usually easier for complex operations like nested radicals. Use them when applying exponent rules, then convert back to radical form for the final answer if needed.

Why does $(x^2)^{1/4}$ equal $x^{1/2}$ and not just $x$ ?

Remember the power rule: $(a^m)^n = a^{mn}$ . So $(x^2)^{1/4} = x^{2 \times 1/4} = x^{1/2}$ . We multiply the exponents, not add them!

Can I check my answer by substituting a value?

Yes! Try $x = 4$ : $\sqrt{\sqrt{3 \cdot 16}} = \sqrt{\sqrt{48}} ≈ 2.63$ , and $\sqrt[4]{3}\cdot\sqrt{4} = 1.316 \times 2 ≈ 2.63$ ✓

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Simplify the Nested Square Root: √(√(3x²))

Nested Radicals with Exponent Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just simplify the inner square root first?

What's the difference between $\sqrt[4]{3}\cdot\sqrt{x}$ and $x\sqrt[4]{3}$ ?

How do I know when to use fractional exponents versus radicals?

Why does $(x^2)^{1/4}$ equal $x^{1/2}$ and not just $x$ ?

Can I check my answer by substituting a value?

🌟 Unlock Your Math Potential

More Questions

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Simplify the Nested Square Root: √(√(3x²))

Nested Radicals with Exponent Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just simplify the inner square root first?

What's the difference between 34⋅x\sqrt[4]{3}\cdot\sqrt{x}43​⋅x​ and x34x\sqrt[4]{3}x43​?

How do I know when to use fractional exponents versus radicals?

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Can I check my answer by substituting a value?

🌟 Unlock Your Math Potential

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Simplify the Nested Radical: Sixth Root of Square Root of x^12

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

What's the difference between $\sqrt[4]{3}\cdot\sqrt{x}$ and $x\sqrt[4]{3}$ ?

Why does $(x^2)^{1/4}$ equal $x^{1/2}$ and not just $x$ ?