Solve the Nested Radical: Cube Root of Square Root of 729

Q: Why convert radicals to exponent form?

Converting radicals to fractional exponents lets you use the power rule $ (a^m)^n = a^{m \cdot n} $. This makes nested radicals much easier to simplify!

Q: How do I know what power 729 is?

Look for perfect powers by trying small bases. Since $ 3^3 = 27 $ and $ 3^6 = 729 $, you can recognize that $ 729 = 3^6 $.

Q: What if I can't find the perfect power?

If the base isn't a perfect power, you might need to factor it completely into prime factors, then group them to find the highest power that works.

Q: Can I solve this by working from inside out?

Yes, but it's less efficient! Computing $ \sqrt{729} = 27 $ first, then $ \sqrt[3]{27} = 3 $ works, but the exponent method is faster and less error-prone.

Q: What does the fraction 1/6 mean as an exponent?

The exponent $ \frac{1}{6} $ means sixth root! So $ 729^{\frac{1}{6}} = \sqrt[6]{729} $, which equals 3 because $ 3^6 = 729 $.

Nested Radicals with Exponent Laws

Solve the following exercise:

$\sqrt[3]{\sqrt{729}}=$

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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 (X) in a root of the order (B) in a root of the order (A)

00:14 The result equals the number (X) in a root of the order of their product (A times B)

00:20 Apply this formula to our exercise

00:28 Calculate the order multiplication

00:39 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:

$\sqrt[3]{\sqrt{729}}=$

Step-by-step solution

To solve this problem, we need to evaluate $\sqrt[3]{\sqrt{729}}$ .

Step 1: Convert the expression into exponent form. We know $\sqrt{729} = 729^{\frac{1}{2}}$ . Thus, $\sqrt[3]{\sqrt{729}} = (729^{\frac{1}{2}})^{\frac{1}{3}}$ .

Step 2: Apply the formula for exponents $(a^m)^n = a^{m \cdot n}$ . Thus, $(729^{\frac{1}{2}})^{\frac{1}{3}} = 729^{\frac{1}{2 \cdot 1/3}} = 729^{\frac{1}{6}}$ .

Step 3: Find the base of 729 as a power of an integer. Observing, 729 = $3^6$ because $3^6 = 729$ .

Step 4: Substitute the power of the base: $729^{\frac{1}{6}} = (3^6)^{\frac{1}{6}} = 3^{6 \cdot \frac{1}{6}} = 3^1 = 3$ .

Therefore, the solution to the problem is $\sqrt[3]{\sqrt{729}} = 3$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: Convert radicals to fractional exponents before combining
Technique: $(a^m)^n = a^{m \cdot n}$ , so $(729^{\frac{1}{2}})^{\frac{1}{3}} = 729^{\frac{1}{6}}$
Check: Prime factorize the base: $729 = 3^6$ , so $3^6 \cdot \frac{1}{6} = 3^1 = 3$ ✓

Common Mistakes

Avoid these frequent errors

Computing each radical separately from inside out
Don't calculate $\sqrt{729} = 27$ first, then $\sqrt[3]{27} = 3$ ! While this gives the right answer, it's inefficient and can lead to calculation errors with larger numbers. Always convert to exponent form first: $\sqrt[3]{\sqrt{729}} = 729^{\frac{1}{6}}$ .

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 convert radicals to exponent form?

Converting radicals to fractional exponents lets you use the power rule $(a^m)^n = a^{m \cdot n}$ . This makes nested radicals much easier to simplify!

How do I know what power 729 is?

Look for perfect powers by trying small bases. Since $3^3 = 27$ and $3^6 = 729$ , you can recognize that $729 = 3^6$ .

What if I can't find the perfect power?

If the base isn't a perfect power, you might need to factor it completely into prime factors, then group them to find the highest power that works.

Can I solve this by working from inside out?

Yes, but it's less efficient! Computing $\sqrt{729} = 27$ first, then $\sqrt[3]{27} = 3$ works, but the exponent method is faster and less error-prone.

What does the fraction 1/6 mean as an exponent?

The exponent $\frac{1}{6}$ means sixth root! So $729^{\frac{1}{6}} = \sqrt[6]{729}$ , which equals 3 because $3^6 = 729$ .

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