Simplify the Expression: √5 ÷ ⁴√5 Step-by-Step

Q: Why can't I just cancel out the 5s in the radicals?

You cannot simply cancel the 5s because they have different radical indices. $ \sqrt{5} $ and $ \sqrt[4]{5} $ are fundamentally different expressions that must be converted to exponent form first.

Q: How do I remember which exponent goes with which radical?

Remember: the index becomes the denominator! So $ \sqrt{5} = 5^{1/2} $ (index 2) and $ \sqrt[4]{5} = 5^{1/4} $ (index 4). The numerator is always 1 for simple radicals.

Q: Why do I subtract the exponents when dividing?

This follows the quotient rule for exponents: $ \frac{a^m}{a^n} = a^{m-n} $. When you divide powers with the same base, you subtract the exponents!

Q: How do I subtract fractions like 1/2 - 1/4?

Find a common denominator! Convert $ \frac{1}{2} $ to $ \frac{2}{4} $, then subtract: $ \frac{2}{4} - \frac{1}{4} = \frac{1}{4} $.

Q: Can I leave my answer as 5^(1/4) instead of converting back?

Both forms are correct! $ 5^{1/4} $ and $ \sqrt[4]{5} $ are equivalent. However, most problems expect the radical form as the final answer.

Exponent Rules with Radical Division

Solve the following exercise:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem. Ready?

00:09 Every number here is raised to the power of one.

00:14 And remember, a regular root means the second root.

00:19 If you have a root of order B of a number X, to the power A:

00:24 It equals number X, raised to the power of A divided by B.

00:29 Let's apply this formula to our problem. Try it!

00:33 Dividing powers with the same base. A, divided by B.

00:40 It's the common base to the power of A minus B.

00:44 Use this formula. Subtract the powers.

00:49 Find the common denominator and calculate the power.

01:04 Now, apply it in the reverse. Take your time.

01:08 Convert from power to the fourth root.

01:11 And that's the solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

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

Step-by-step solution

Let's simplify the expression $\frac{\sqrt{5}}{\sqrt[4]{5}}$ using the rules of exponents:

First, we convert $\sqrt{5}$ to exponent form: $\sqrt{5} = 5^{1/2}$ .
Next, we convert $\sqrt[4]{5}$ to exponent form: $\sqrt[4]{5} = 5^{1/4}$ .
Now, divide the two expressions: $\frac{5^{1/2}}{5^{1/4}} = 5^{1/2 - 1/4}$ .
Subtract the exponents: $5^{1/2 - 1/4} = 5^{2/4 - 1/4} = 5^{1/4}$ .

The problem is simplified to $\sqrt[4]{5}$ .

Therefore, the simplified form of the given expression is $\sqrt[4]{5}$ .

Final Answer

$\sqrt[4]{5}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert radicals to fractional exponents before dividing
Technique: $\sqrt{5} = 5^{1/2}$ and $\sqrt[4]{5} = 5^{1/4}$
Check: Verify $5^{1/2-1/4} = 5^{1/4} = \sqrt[4]{5}$ ✓

Common Mistakes

Avoid these frequent errors

Attempting to divide radicals directly without converting to exponents
Don't try to divide $\sqrt{5} ÷ \sqrt[4]{5}$ as radicals = confusion and wrong answers! Direct radical division is complicated and error-prone. Always convert to fractional exponents first, then use subtraction rule.

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 can't I just cancel out the 5s in the radicals?

You cannot simply cancel the 5s because they have different radical indices. $\sqrt{5}$ and $\sqrt[4]{5}$ are fundamentally different expressions that must be converted to exponent form first.

How do I remember which exponent goes with which radical?

Remember: the index becomes the denominator! So $\sqrt{5} = 5^{1/2}$ (index 2) and $\sqrt[4]{5} = 5^{1/4}$ (index 4). The numerator is always 1 for simple radicals.

Why do I subtract the exponents when dividing?

This follows the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$ . When you divide powers with the same base, you subtract the exponents!

How do I subtract fractions like 1/2 - 1/4?

Find a common denominator! Convert $\frac{1}{2}$ to $\frac{2}{4}$ , then subtract: $\frac{2}{4} - \frac{1}{4} = \frac{1}{4}$ .

Can I leave my answer as 5^(1/4) instead of converting back?

Both forms are correct! $5^{1/4}$ and $\sqrt[4]{5}$ are equivalent. However, most problems expect the radical form as the final answer.

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