Simplify the Fraction: Fourth Root of 3 Divided by Sixth Root of 3

Q: How do I convert roots to exponents?

Use the rule: nth root of a = a^(1/n). So $ \sqrt[4]{3} = 3^{1/4} $ and $ \sqrt[6]{3} = 3^{1/6} $.

Q: Why can't I just divide 4 by 6?

The root indices (4 and 6) become denominators in the exponents, not the whole exponent! You need to subtract $ \frac{1}{4} - \frac{1}{6} $, not divide 4 ÷ 6.

Q: How do I subtract fractions with different denominators?

Find the LCD (Least Common Multiple) of the denominators. For 4 and 6, the LCD is 12: $ \frac{1}{4} = \frac{3}{12} $ and $ \frac{1}{6} = \frac{2}{12} $.

Q: Can I leave my answer as a root instead?

Yes! $ 3^{1/12} $ can also be written as $ \sqrt[12]{3} $. Both forms are correct, but the fractional exponent form is often preferred.

Q: What if the bases were different numbers?

You can only use the quotient rule $ \frac{a^m}{a^n} = a^{m-n} $ when the bases are the same. If bases differ, you'd need to calculate each root separately first.

Exponent Rules with Fractional Powers

Solve the following exercise:

$\frac{\sqrt[4]{3}}{\sqrt[6]{3}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 Every number is to the power of 1

00:09 When we have a root of the order (B) on a number (X) to the power of (A)

00:14 The result equals number (X) to the power of (A divided by B)

00:19 Apply this formula to our exercise

00:26 When we have division of powers (A/B) with equal bases

00:31 The result equals the common base to the power of the difference of exponents (A - B)

00:36 Apply this formula to our exercise, and subtract between the powers

00:43 Determine the common denominator and proceed to calculate the power

00:50 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\frac{\sqrt[4]{3}}{\sqrt[6]{3}}=$

Step-by-step solution

To solve this problem, we must simplify the expression $\frac{\sqrt[4]{3}}{\sqrt[6]{3}}$ . We will follow these steps:

Step 1: Convert roots to exponents.
$\sqrt[4]{3}$ is equivalent to $3^{\frac{1}{4}}$ , and $\sqrt[6]{3}$ is equivalent to $3^{\frac{1}{6}}$ .
Step 2: Apply the quotient of powers formula.
Using the property $\frac{a^m}{a^n} = a^{m-n}$ , we have: $\frac{3^{\frac{1}{4}}}{3^{\frac{1}{6}}} = 3^{\frac{1}{4} - \frac{1}{6}}$
Step 3: Perform the subtraction of the exponents.
To subtract the fractions $\frac{1}{4}$ and $\frac{1}{6}$ , find a common denominator. The least common multiple of 4 and 6 is 12, so: $\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$ Thus, $\frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}$
Step 4: Simplify the result.
Thus, we have: $3^{\frac{1}{4} - \frac{1}{6}} = 3^{\frac{1}{12}}$

Hence, the simplified form of the given expression is $3^{\frac{1}{12}}$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Rule: Convert roots to fractional exponents before dividing
Technique: Subtract exponents: $3^{1/4} ÷ 3^{1/6} = 3^{1/4-1/6}$
Check: Verify common denominators: 3/12 - 2/12 = 1/12 ✓

Common Mistakes

Avoid these frequent errors

Converting roots incorrectly or forgetting exponent rules
Don't try to divide the roots directly or convert $\sqrt[4]{3}$ to $3^4$ ! This gives completely wrong answers like $3^{12}$ . Always convert nth root to the exponent 1/n first, then apply the quotient rule.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that is equal to the following:

$\sqrt{a}:\sqrt{b}$

FAQ

Everything you need to know about this question

How do I convert roots to exponents?

Use the rule: nth root of a = a^(1/n). So $\sqrt[4]{3} = 3^{1/4}$ and $\sqrt[6]{3} = 3^{1/6}$ .

Why can't I just divide 4 by 6?

The root indices (4 and 6) become denominators in the exponents, not the whole exponent! You need to subtract $\frac{1}{4} - \frac{1}{6}$ , not divide 4 ÷ 6.

How do I subtract fractions with different denominators?

Find the LCD (Least Common Multiple) of the denominators. For 4 and 6, the LCD is 12: $\frac{1}{4} = \frac{3}{12}$ and $\frac{1}{6} = \frac{2}{12}$ .

Can I leave my answer as a root instead?

Yes! $3^{1/12}$ can also be written as $\sqrt[12]{3}$ . Both forms are correct, but the fractional exponent form is often preferred.

What if the bases were different numbers?

You can only use the quotient rule $\frac{a^m}{a^n} = a^{m-n}$ when the bases are the same. If bases differ, you'd need to calculate each root separately first.

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