Solve: Division of Cube Root and Fourth Root of 5

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

Converting radicals to fractional exponents lets you use the quotient rule! $ \sqrt[3]{5} = 5^{1/3} $ and $ \sqrt[4]{5} = 5^{1/4} $ are much easier to divide using exponent rules.

Q: How do I subtract fractions like 1/3 - 1/4?

Find the common denominator: $ \frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12} $. The LCD of 3 and 4 is 12!

Q: Are both radical and exponential forms correct?

Yes! $ 5^{1/12} $ and $ \sqrt[12]{5} $ represent the exact same number. They're just different ways to write it - like saying one-half or 0.5.

Q: What if the bases were different numbers?

The quotient rule $ \frac{a^m}{a^n} = a^{m-n} $ only works when the bases are the same. If you had $ \frac{\sqrt[3]{5}}{\sqrt[4]{2}} $, you couldn't simplify it this way.

Q: Can I use a calculator to check my answer?

Absolutely! Calculate $ \sqrt[3]{5} ÷ \sqrt[4]{5} $ and compare it to $ \sqrt[12]{5} $. Both should give you approximately 1.145.

Exponent Rules with Radical Division

Solve the following exercise:

$\frac{\sqrt[3]{5}}{\sqrt[4]{5}}=$

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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

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

00:15 The result equals number (X) to the power of (A divided by B)

00:20 Apply this formula to our exercise

00:27 When we have division of powers (A/B) with equal bases

00:36 The result equals the common base to the power of the difference of powers (A - B)

00:39 Apply this formula to our exercise, and subtract between the powers

00:49 Identify the common denominator and calculate the power

00:58 Apply the formula again and convert the power to the 12th root

01:05 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:

$\frac{\sqrt[3]{5}}{\sqrt[4]{5}}=$

Step-by-step solution

To solve the given problem, let us proceed step by step:

Step 1: Convert the given roots into fractional exponents:
The cube root $\sqrt[3]{5}$ can be expressed as $5^{1/3}$ .
The fourth root $\sqrt[4]{5}$ can be expressed as $5^{1/4}$ .
Step 2: Apply the quotient rule for exponents:
For division, $\frac{a^m}{a^n} = a^{m-n}$ .
Thus, $\frac{5^{1/3}}{5^{1/4}} = 5^{1/3 - 1/4}$ .
Step 3: Perform the subtraction of the exponents:
$1/3 - 1/4 = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$ .
Therefore, we have $5^{\frac{1}{12}}$ .
Step 4: Compare with answer choices:
- The expression $5^{\frac{1}{12}}$ directly matches Choice 3.
- Alternatively, we can express this as a 12th root: $\sqrt[12]{5}$ , which matches Choice 1.

Both answer choices (a) $\sqrt[12]{5}$ and (c) $5^{\frac{1}{12}}$ correctly represent the simplified form of the expression.

Thus, the correct solution to the problem is given by Answers (a) and (c).

Final Answer

Answers (a) and (c)

Key Points to Remember

Essential concepts to master this topic

Rule: Convert radicals to fractional exponents before dividing
Technique: $\frac{5^{1/3}}{5^{1/4}} = 5^{1/3 - 1/4} = 5^{1/12}$
Check: Convert back to radical form: $5^{1/12} = \sqrt[12]{5}$ ✓

Common Mistakes

Avoid these frequent errors

Dividing the root indices instead of subtracting exponents
Don't divide 3 ÷ 4 to get $\sqrt[3/4]{5}$ = nonsense! This ignores the quotient rule completely. Always convert to fractional exponents first, then subtract: $5^{1/3 - 1/4} = 5^{1/12}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

FAQ

Everything you need to know about this question

Why do I need to convert radicals to exponents?

Converting radicals to fractional exponents lets you use the quotient rule! $\sqrt[3]{5} = 5^{1/3}$ and $\sqrt[4]{5} = 5^{1/4}$ are much easier to divide using exponent rules.

How do I subtract fractions like 1/3 - 1/4?

Find the common denominator: $\frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$ . The LCD of 3 and 4 is 12!

Are both radical and exponential forms correct?

Yes! $5^{1/12}$ and $\sqrt[12]{5}$ represent the exact same number. They're just different ways to write it - like saying one-half or 0.5.

What if the bases were different numbers?

The quotient rule $\frac{a^m}{a^n} = a^{m-n}$ only works when the bases are the same. If you had $\frac{\sqrt[3]{5}}{\sqrt[4]{2}}$ , you couldn't simplify it this way.

Can I use a calculator to check my answer?

Absolutely! Calculate $\sqrt[3]{5} ÷ \sqrt[4]{5}$ and compare it to $\sqrt[12]{5}$ . Both should give you approximately 1.145.

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