Solve the Nested Square Root: Finding √√12

Q: Why can't I just multiply the square roots together?

Nested radicals are different from multiplication! $ \sqrt{\sqrt{12}} $ means "take the square root of the result of taking the square root of 12," not "multiply two square roots together."

Q: How do I remember the exponent rule for nested radicals?

Think of it as stacking the exponents: $ \sqrt{\sqrt{12}} = (12^{1/2})^{1/2} $. When you have a power raised to another power, you multiply the exponents: $ \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} $.

Q: What's the difference between $ \sqrt[4]{12} $ and $ \sqrt{12} $ ?

$ \sqrt[4]{12} $ asks "what number multiplied by itself 4 times gives 12?" while $ \sqrt{12} $ asks "what number multiplied by itself 2 times gives 12?" The fourth root is a smaller number!

Q: Can I simplify $ \sqrt[4]{12} $ further?

Yes! Since $ 12 = 4 \times 3 = 2^2 \times 3 $, you can write $ \sqrt[4]{12} = \sqrt[4]{2^2 \times 3} = \sqrt[4]{2^2} \times \sqrt[4]{3} = 2^{2/4} \times 3^{1/4} = \sqrt{2} \times \sqrt[4]{3} $.

Q: How can I check my answer without a calculator?

If $ \sqrt[4]{12} $ is correct, then $ (\sqrt[4]{12})^4 = 12 $. Also, $ (\sqrt[4]{12})^2 = \sqrt{12} $, so $ \sqrt{\sqrt{12}} = \sqrt[4]{12} $ makes sense!

Nested Radicals with Exponent Laws

Solve the following exercise:

$\sqrt{\sqrt{12}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's solve this problem step by step.

00:09 Remember, a regular root is a square root, or a root of order two.

00:14 If a number A is raised to power B inside a root of order C,

00:19 the result is A raised to power of B multiplied by C. Let's calculate it together.

00:25 We apply this formula to our exercise.

00:29 and calculate the multiplication.

00:32 Great job! We've found the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{\sqrt{12}}=$

Step-by-step solution

In order to solve the following expression $\sqrt{\sqrt{12}}$ , it needs to be simplified using the properties of exponents and roots. Specifically, we apply the rule that states that the square root of a square root can be expressed as a fourth root.

Let's break down this solution step by step:

First, represent the inner $\sqrt{12}$ as a power: $12^{1/2}$ .
Next, take the square root of this result, which involves raising $12^{1/2}$ to the power of $1/2$ again:
$\left(12^{1/2}\right)^{1/2} = 12^{(1/2) \cdot (1/2)} = 12^{1/4}$ .
According to the rules of exponents, raising an exponent to another power results in multiplying the exponents.
This gives us $12^{1/4}$ , which we can write as the fourth root of 12: $\sqrt[4]{12}$ .

In conclusion the simplification of $\sqrt{\sqrt{12}}$ is $\sqrt[4]{12}$ .

Final Answer

$\sqrt[4]{12}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested square roots to fractional exponent form
Technique: Apply power rule: $(a^{m})^n = a^{m \cdot n}$ , so $(12^{1/2})^{1/2} = 12^{1/4}$
Check: Verify that $\sqrt[4]{12} \times \sqrt[4]{12} \times \sqrt[4]{12} \times \sqrt[4]{12} = 12$ ✓

Common Mistakes

Avoid these frequent errors

Treating nested radicals as simple multiplication
Don't think $\sqrt{\sqrt{12}} = \sqrt{12} \times \sqrt{12} = 12$ ! This confuses composition with multiplication and gives the wrong answer. Always convert to fractional exponents: $\sqrt{\sqrt{12}} = (12^{1/2})^{1/2} = 12^{1/4}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

FAQ

Everything you need to know about this question

Why can't I just multiply the square roots together?

Nested radicals are different from multiplication! $\sqrt{\sqrt{12}}$ means "take the square root of the result of taking the square root of 12," not "multiply two square roots together."

How do I remember the exponent rule for nested radicals?

Think of it as stacking the exponents: $\sqrt{\sqrt{12}} = (12^{1/2})^{1/2}$ . When you have a power raised to another power, you multiply the exponents: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ .

What's the difference between $\sqrt[4]{12}$ and $\sqrt{12}$ ?

$\sqrt[4]{12}$ asks "what number multiplied by itself 4 times gives 12?" while $\sqrt{12}$ asks "what number multiplied by itself 2 times gives 12?" The fourth root is a smaller number!

Can I simplify $\sqrt[4]{12}$ further?

Yes! Since $12 = 4 \times 3 = 2^2 \times 3$ , you can write $\sqrt[4]{12} = \sqrt[4]{2^2 \times 3} = \sqrt[4]{2^2} \times \sqrt[4]{3} = 2^{2/4} \times 3^{1/4} = \sqrt{2} \times \sqrt[4]{3}$ .

How can I check my answer without a calculator?

If $\sqrt[4]{12}$ is correct, then $(\sqrt[4]{12})^4 = 12$ . Also, $(\sqrt[4]{12})^2 = \sqrt{12}$ , so $\sqrt{\sqrt{12}} = \sqrt[4]{12}$ makes sense!

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Solve the Nested Square Root: Finding √√12

Nested Radicals with Exponent Laws

❤️ 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 can't I just multiply the square roots together?

How do I remember the exponent rule for nested radicals?

What's the difference between $\sqrt[4]{12}$ and $\sqrt{12}$ ?

Can I simplify $\sqrt[4]{12}$ further?

How can I check my answer without a calculator?

🌟 Unlock Your Math Potential

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Solve the Nested Square Root: Finding √√12

Nested Radicals with Exponent Laws

❤️ 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 can't I just multiply the square roots together?

How do I remember the exponent rule for nested radicals?

What's the difference between 124 \sqrt[4]{12} 412​ and 12 \sqrt{12} 12​?

Can I simplify 124 \sqrt[4]{12} 412​ further?

How can I check my answer without a calculator?

🌟 Unlock Your Math Potential

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Solve the Nested Radical: Fourth Root of Square Root of 6

What's the difference between $\sqrt[4]{12}$ and $\sqrt{12}$ ?

Can I simplify $\sqrt[4]{12}$ further?