Multiply and Simplify: Fifth Root and Sixth Root of √3

Q: Why do I need to convert radicals to exponential form?

Converting makes it easier to use exponent rules! $ \sqrt[n]{a} = a^{1/n} $ lets you work with fractions instead of confusing nested radical symbols.

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

Start from the inside out: $ \sqrt{3} = 3^{1/2} $ first, then apply the outer radical. So $ \sqrt[5]{\sqrt{3}} = (3^{1/2})^{1/5} $.

Q: When do I add exponents versus multiply them?

Add exponents when multiplying powers with the same base: $ a^m \cdot a^n = a^{m+n} $. Multiply exponents when raising a power to a power: $ (a^m)^n = a^{mn} $.

Q: Do I need to simplify the fraction in the exponent?

The answer $ 3^{\frac{1}{10}+\frac{1}{12}} $ is correct as shown. You could add the fractions to get $ 3^{\frac{11}{60}} $, but both forms are acceptable.

Q: What if the bases were different numbers?

If you had different bases like $ \sqrt[5]{\sqrt{2}} \cdot \sqrt[6]{\sqrt{3}} $, you couldn't combine them easily. The same base (3 in this problem) is what allows us to add exponents.

Radical Expressions with Nested Roots

Complete the following exercise:

$\sqrt[5]{\sqrt{3}}\cdot\sqrt[6]{\sqrt{3}}=$

❤️ 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

00:05 A "regular" root is of the order 2

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

00:17 The result equals the root of the multiplication of the orders

00:21 Apply this formula to our exercise

00:31 Let's calculate the multiplication of orders

00:37 When we have a root of order (C) on number (A) to the power of (B)

00:40 The result equals number (A) to the power of (B divided by C)

00:44 Apply this formula to our exercise, where each number is to the power of 1

00:56 When we have multiplication between powers with equal bases

00:59 The result equals the base with power equal to the sum of the powers

01:04 Apply this formula to our exercise

01:07 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:

$\sqrt[5]{\sqrt{3}}\cdot\sqrt[6]{\sqrt{3}}=$

Step-by-step solution

To simplify the expression $\sqrt[5]{\sqrt{3}} \cdot \sqrt[6]{\sqrt{3}}$ , follow these steps:

Step 1: Represent $\sqrt{3}$ as $3^{1/2}$ because $\sqrt{3} = 3^{1/2}$ .
Step 2: Express $\sqrt[5]{\sqrt{3}}$ in exponential form: $\sqrt[5]{\sqrt{3}} = (\sqrt{3})^{1/5} = (3^{1/2})^{1/5} = 3^{(1/2) \times (1/5)} = 3^{1/10}$ .
Step 3: Express $\sqrt[6]{\sqrt{3}}$ in exponential form: $\sqrt[6]{\sqrt{3}} = (\sqrt{3})^{1/6} = (3^{1/2})^{1/6} = 3^{(1/2) \times (1/6)} = 3^{1/12}$ .
Step 4: Multiply the two expressions using properties of exponents: $3^{1/10} \cdot 3^{1/12} = 3^{(1/10 + 1/12)}$ .

Therefore, the simplified expression is $3^{\frac{1}{10}+\frac{1}{12}}$ .

Final Answer

$3^{\frac{1}{10}+\frac{1}{12}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested radicals to exponential form using $a^{m/n}$
Technique: $\sqrt[5]{\sqrt{3}} = (3^{1/2})^{1/5} = 3^{1/10}$
Check: Same base allows adding exponents: $3^{1/10} \cdot 3^{1/12} = 3^{(1/10 + 1/12)}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the radical indices instead of adding exponents
Don't multiply 5 × 6 = 30 to get $\sqrt[30]{3}$ ! This treats the problem like nested radicals are being combined incorrectly. Always convert to exponential form first, then add the exponents when multiplying powers with the same base.

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 radicals to exponential form?

Converting makes it easier to use exponent rules! $\sqrt[n]{a} = a^{1/n}$ lets you work with fractions instead of confusing nested radical symbols.

How do I handle the nested square root inside?

Start from the inside out: $\sqrt{3} = 3^{1/2}$ first, then apply the outer radical. So $\sqrt[5]{\sqrt{3}} = (3^{1/2})^{1/5}$ .

When do I add exponents versus multiply them?

Add exponents when multiplying powers with the same base: $a^m \cdot a^n = a^{m+n}$ . Multiply exponents when raising a power to a power: $(a^m)^n = a^{mn}$ .

Do I need to simplify the fraction in the exponent?

The answer $3^{\frac{1}{10}+\frac{1}{12}}$ is correct as shown. You could add the fractions to get $3^{\frac{11}{60}}$ , but both forms are acceptable.

What if the bases were different numbers?

If you had different bases like $\sqrt[5]{\sqrt{2}} \cdot \sqrt[6]{\sqrt{3}}$ , you couldn't combine them easily. The same base (3 in this problem) is what allows us to add exponents.

🌟 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

Multiply and Simplify: Fifth Root and Sixth Root of √3

Radical Expressions with Nested Roots

❤️ 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 radicals to exponential form?

How do I handle the nested square root inside?

When do I add exponents versus multiply them?

Do I need to simplify the fraction in the exponent?

What if the bases were different numbers?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

Simplify the Expression: Cube Root of Square Root of 64 × Cube Root of 64

Solve Nested Square Roots: √√4 × √√2 Multiplication Problem

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

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

Root of a Root

Simplify √25 · ∛(√25): Multiple Radical Expression Problem

Solve Nested Square Roots: √√49 × √√16 Multiplication Problem

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

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