Solve Nested Radicals: Cube Root of Square Root of 144

Q: Why can't I just calculate the square root of 144 first?

While $ \sqrt{144} = 12 $ is correct, finding $ \sqrt[3]{12} $ gives you a messy decimal. The exponent rule method keeps everything in exact radical form, which is usually preferred!

Q: How do I multiply fractional exponents?

Just multiply the fractions! For $ \frac{1}{2} \times \frac{1}{3} $, multiply numerators: $ 1 \times 1 = 1 $, and denominators: $ 2 \times 3 = 6 $, giving $ \frac{1}{6} $.

Q: What does the sixth root of 144 actually equal?

Since $ 2^6 = 64 $ and $ 3^6 = 729 $, we know $ \sqrt[6]{144} $ is between 2 and 3. The exact answer $ \sqrt[6]{144} $ is preferred over the decimal approximation.

Q: When should I use fractional exponents vs. radical notation?

Use fractional exponents when combining operations like this problem. Use radical notation for your final answer since it's often clearer to read.

Q: Can this method work for other nested radicals?

Absolutely! For any nested radical like $ \sqrt[n]{\sqrt[m]{a}} $, convert to $ (a^{1/m})^{1/n} = a^{1/(m \times n)} = \sqrt[mn]{a} $.

Nested Radicals with Exponent Laws

Solve the following exercise:

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

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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 raised to the second power

00:08 When we have a number (A) in a root raised to (B) in a root raised to (C)

00:16 The result equals number (A) to the power of their quotient (B divided by C)

00:21 Let's apply this formula to our exercise

00:25 Calculate the order multiplication

00:33 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:

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

Step-by-step solution

To solve this problem, let's follow these steps:

Express the square root as a fractional exponent.
Express the cube root as another fractional exponent.
Multiply the exponents together using the rule $(a^m)^n = a^{m \times n}$ .
Recapture the result as a root expression.

Let's apply these steps:
Step 1: The square root of 144 can be expressed as $144^{1/2}$ .
Step 2: We need the cube root of this expression, so we have $(144^{1/2})^{1/3}$ .
Step 3: Using the property of exponents $(a^m)^n = a^{m \times n}$ , we multiply the exponents: $(144^{1/2})^{1/3} = 144^{(1/2) \times (1/3)} = 144^{1/6}$ .
Step 4: Re-express this as a root: Since $144^{1/6}$ is equivalent to the sixth root, we have $\sqrt[6]{144}$ .

Therefore, the solution to the problem is $\sqrt[6]{144}$ , which corresponds to choice 3.

Final Answer

$\sqrt[6]{144}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert radicals to fractional exponents before combining operations
Technique: $\sqrt[3]{\sqrt{144}} = (144^{1/2})^{1/3} = 144^{1/6}$
Check: Verify $144^{1/6} = \sqrt[6]{144}$ using root notation ✓

Common Mistakes

Avoid these frequent errors

Trying to simplify the inner radical first
Don't calculate $\sqrt{144} = 12$ first and then find $\sqrt[3]{12}$ = wrong approach! This makes the problem harder and often leads to decimal approximations. Always use the exponent multiplication rule $(a^m)^n = a^{m \times n}$ to combine the operations algebraically.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\sqrt{4}}=$

FAQ

Everything you need to know about this question

Why can't I just calculate the square root of 144 first?

While $\sqrt{144} = 12$ is correct, finding $\sqrt[3]{12}$ gives you a messy decimal. The exponent rule method keeps everything in exact radical form, which is usually preferred!

How do I multiply fractional exponents?

Just multiply the fractions! For $\frac{1}{2} \times \frac{1}{3}$ , multiply numerators: $1 \times 1 = 1$ , and denominators: $2 \times 3 = 6$ , giving $\frac{1}{6}$ .

What does the sixth root of 144 actually equal?

Since $2^6 = 64$ and $3^6 = 729$ , we know $\sqrt[6]{144}$ is between 2 and 3. The exact answer $\sqrt[6]{144}$ is preferred over the decimal approximation.

When should I use fractional exponents vs. radical notation?

Use fractional exponents when combining operations like this problem. Use radical notation for your final answer since it's often clearer to read.

Can this method work for other nested radicals?

Absolutely! For any nested radical like $\sqrt[n]{\sqrt[m]{a}}$ , convert to $(a^{1/m})^{1/n} = a^{1/(m \times n)} = \sqrt[mn]{a}$ .

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