Simplify the Nested Radical: Cube Root of Square Root of 144x³

Q: Why do we convert radicals to fractional exponents?

Converting to fractional exponents makes nested radicals much easier! Instead of dealing with $\sqrt[3]{\sqrt{...}}$, you can use simple exponent rules like $(a^m)^n = a^{mn}$.

Q: How do I know when to separate the variables and constants?

You can separate them when you have a product inside the radical. Use the property $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ to split $(144x^3)^{1/6}$ into $144^{1/6} \cdot (x^3)^{1/6}$.

Q: What if I get different radical indices in my final answer?

That's normal! In this problem, we get $\sqrt[6]{144} \cdot \sqrt{x}$ with different indices (6th root and square root). Don't try to force them to match - this is the simplest form.

Q: Can I simplify the number under the radical further?

Sometimes! For $\sqrt[6]{144}$, you could factor 144, but since 144 is not a perfect 6th power, it's usually best to leave it as is unless specifically asked to factor.

Q: How can I check if my answer is correct?

Work backwards! Raise your answer to the 6th power: $(\sqrt[6]{144} \cdot \sqrt{x})^6 = 144 \cdot x^3 = 144x^3$. This should equal what's inside the original nested radical.

Nested Radicals with Exponent Rules

Complete the following exercise:

$\sqrt[3]{\sqrt{144x^3}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 A "regular" root is of the order 2

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

00:15 The result equals the number (A) to the power of their product (B times C)

00:19 We'll apply this formula to our exercise

00:25 When we have a root of the multiplication (A times B)

00:28 It can be written as multiplication of the root of each term

00:33 We'll apply this formula to our exercise, and proceed to break down the root

00:44 When we have a number (A) to the power of (B) in a root of order (C)

00:47 The result equals the number (A) to the power of their quotient (B divided by C)

00:52 We'll apply this formula to our exercise, and proceed to calculate the power quotient

01:05 We'll apply this formula again in the reverse direction, converting from power to root

01:18 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:

$\sqrt[3]{\sqrt{144x^3}}=$

Step-by-step solution

To solve the expression $\sqrt[3]{\sqrt{144x^3}}$ , let's proceed step by step:

Step 1: Express the inner square root using fractional exponents:
$\sqrt{144x^3} = (144x^3)^{1/2}$
Step 2: Apply the cube root to the expression obtained in Step 1:
$\sqrt[3]{(144x^3)^{1/2}} = ((144x^3)^{1/2})^{1/3}$
Step 3: Use the exponent rule $(a^m)^n = a^{m \cdot n}$ :
$((144x^3)^{1/2})^{1/3} = (144x^3)^{(1/2) \cdot (1/3)} = (144x^3)^{1/6}$
Step 4: Separate the powers for 144 and $x^3$ :
$(144)^{1/6} \cdot (x^3)^{1/6} = \sqrt[6]{144} \cdot x^{3/6} = \sqrt[6]{144} \cdot x^{1/2} = \sqrt[6]{144} \cdot \sqrt{x}$

Therefore, the simplified expression is $\sqrt[6]{144} \cdot \sqrt{x}$ , which corresponds to the third choice.

Final Answer

$\sqrt[6]{144}\cdot\sqrt{x}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested radicals to fractional exponents for easier simplification
Technique: Use $(a^m)^n = a^{mn}$ to combine: $((144x^3)^{1/2})^{1/3} = (144x^3)^{1/6}$
Check: Verify by working backwards: $(\sqrt[6]{144} \cdot \sqrt{x})^6 = 144x^3$ ✓

Common Mistakes

Avoid these frequent errors

Trying to simplify inner radical before applying outer radical
Don't simplify $\sqrt{144x^3} = 12x\sqrt{x}$ first then take cube root = complicated fractions! This creates unnecessary complexity with mixed radicals and fractional exponents. Always convert to fractional exponents immediately and use exponent rules.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

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

FAQ

Everything you need to know about this question

Why do we convert radicals to fractional exponents?

Converting to fractional exponents makes nested radicals much easier! Instead of dealing with $\sqrt[3]{\sqrt{...}}$ , you can use simple exponent rules like $(a^m)^n = a^{mn}$ .

How do I know when to separate the variables and constants?

You can separate them when you have a product inside the radical. Use the property $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ to split $(144x^3)^{1/6}$ into $144^{1/6} \cdot (x^3)^{1/6}$ .

What if I get different radical indices in my final answer?

That's normal! In this problem, we get $\sqrt[6]{144} \cdot \sqrt{x}$ with different indices (6th root and square root). Don't try to force them to match - this is the simplest form.

Can I simplify the number under the radical further?

Sometimes! For $\sqrt[6]{144}$ , you could factor 144, but since 144 is not a perfect 6th power, it's usually best to leave it as is unless specifically asked to factor.

How can I check if my answer is correct?

Work backwards! Raise your answer to the 6th power: $(\sqrt[6]{144} \cdot \sqrt{x})^6 = 144 \cdot x^3 = 144x^3$ . This should equal what's inside the original nested radical.

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