Solve: Multiplication of Cube Roots with Nested Square Roots (√[3]{√3}·√[3]{√4})

Q: Why do I need to convert roots to exponents?

Converting to exponents makes multiplication much easier! With roots like $ \sqrt[3]{\sqrt{3}} $, it's hard to see what to do next. But as $ 3^{1/6} $, you can clearly apply exponent rules.

Q: How do I handle the nested square root inside the cube root?

Work from the inside out! First, $ \sqrt{3} = 3^{1/2} $, then $ \sqrt[3]{3^{1/2}} = (3^{1/2})^{1/3} = 3^{1/6} $ using the power rule.

Q: What's the exponent rule for multiplying same powers?

When multiplying expressions with the same exponent, you multiply the bases: $ a^n \cdot b^n = (a \cdot b)^n $. So $ 3^{1/6} \cdot 4^{1/6} = (3 \cdot 4)^{1/6} = 12^{1/6} $.

Q: How do I convert the final answer back to root form?

Remember that fractional exponents become roots! The denominator of the fraction becomes the root index: $ 12^{1/6} = \sqrt[6]{12} $.

Nested Root Simplification with Exponential Properties

Complete the following exercise:

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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 raised to the second power

00:13 When there is a root of order (C) of root (B)

00:16 The result equals the root of the product of the orders

00:21 Apply this formula to our exercise

00:36 When we have a product of 2 numbers (A and B) in a root of order (C)

00:39 The result equals their product (A times B) in a root of order (C)

00:45 Apply this formula to our exercise

00:51 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:

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

Step-by-step solution

To solve $\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{4}}$ , we will convert these expressions into powers:

Step 1: Express each root as a power:
$\sqrt[3]{\sqrt{3}} = (\sqrt{3})^{1/3} = 3^{1/6}$
$\sqrt[3]{\sqrt{4}} = (\sqrt{4})^{1/3} = 4^{1/6}$

Step 2: Multiply the expressions using the property of exponents:
$3^{1/6} \cdot 4^{1/6} = (3 \cdot 4)^{1/6} = 12^{1/6}$

Therefore, the simplified expression is $\sqrt[6]{12}$ .

Final Answer

$\sqrt[6]{12}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested roots to fractional exponents for easier manipulation
Technique: $\sqrt[3]{\sqrt{3}} = 3^{1/6}$ using exponent chain rule
Check: Verify $12^{1/6} = \sqrt[6]{12}$ by converting back ✓

Common Mistakes

Avoid these frequent errors

Adding cube root exponents instead of multiplying bases
Don't add the exponents like $3^{1/6} \cdot 4^{1/6} = 7^{1/6}$ = wrong answer! This treats the bases as if they're being added. Always multiply the bases when exponents are the same: $3^{1/6} \cdot 4^{1/6} = (3 \cdot 4)^{1/6}$ .

Practice Quiz

Test your knowledge with interactive questions

Choose the largest value

FAQ

Everything you need to know about this question

Why do I need to convert roots to exponents?

Converting to exponents makes multiplication much easier! With roots like $\sqrt[3]{\sqrt{3}}$ , it's hard to see what to do next. But as $3^{1/6}$ , you can clearly apply exponent rules.

How do I handle the nested square root inside the cube root?

Work from the inside out! First, $\sqrt{3} = 3^{1/2}$ , then $\sqrt[3]{3^{1/2}} = (3^{1/2})^{1/3} = 3^{1/6}$ using the power rule.

What's the exponent rule for multiplying same powers?

When multiplying expressions with the same exponent, you multiply the bases: $a^n \cdot b^n = (a \cdot b)^n$ . So $3^{1/6} \cdot 4^{1/6} = (3 \cdot 4)^{1/6} = 12^{1/6}$ .

How do I convert the final answer back to root form?

Remember that fractional exponents become roots! The denominator of the fraction becomes the root index: $12^{1/6} = \sqrt[6]{12}$ .

Why isn't the answer just 12?

Be careful! We're not just multiplying 3 × 4. We're multiplying nested roots, which requires converting to fractional exponents first. The final exponent $1/6$ means we need the 6th root of 12.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rules of Roots questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Solve: Multiplication of Cube Roots with Nested Square Roots (√[3]{√3}·√[3]{√4})

Nested Root Simplification with Exponential Properties

❤️ 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 do I need to convert roots to exponents?

How do I handle the nested square root inside the cube root?

What's the exponent rule for multiplying same powers?

How do I convert the final answer back to root form?

Why isn't the answer just 12?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

Multiply and Simplify: (⁵√√3) × (⁵√√3) Radical Expression

Solve Nested Square Roots: √√16 × √√8 Multiplication Problem

Solve: (√36)/(√2·√3) + (√25)/5 | Adding Fractions with Square Roots

Simplify the Radical Expression: (√10 × √30)/√100

Root of a Root

Solve Nested Radicals: Sixth Root of √64 × Fourth Root of ∛16

Solve Nested Radicals: Cube Root of Square Root of 16 × Square Root Problems

Solve Nested Square Roots: √√2 × √√4 Product Evaluation

Maximum Value Identification: Comparing Numerical Quantities