Finding the Largest Value: Numerical Comparison Exercise

Q: Why do we convert radicals to fractional exponents?

Converting to fractional exponents makes comparison much easier! When you have the same base (like 4), you just compare the exponents: $ \frac{1}{2} > \frac{1}{6} > \frac{1}{12} $.

Q: How do I handle nested radicals like √(∛4)?

Work from inside out! First convert $ \sqrt[3]{4} = 4^{\frac{1}{3}} $, then $ \sqrt{4^{\frac{1}{3}}} = (4^{\frac{1}{3}})^{\frac{1}{2}} = 4^{\frac{1}{6}} $.

Q: What's the rule for multiplying exponents in nested radicals?

When you have nested operations, multiply the exponents! For $ \sqrt{\sqrt[6]{4}} $, you get $ 4^{\frac{1}{6} \times \frac{1}{2}} = 4^{\frac{1}{12}} $.

Q: Why is √4 the largest value here?

Because $ \sqrt{4} = 4^{\frac{1}{2}} $ has the largest exponent! Since $ \frac{1}{2} = \frac{6}{12} $ is bigger than $ \frac{1}{6} = \frac{2}{12} $ and $ \frac{1}{12} $.

Q: Can I just calculate decimal values instead?

Yes, but it's less efficient! You'd need a calculator and might lose accuracy. The fractional exponent method is faster and shows the mathematical relationship clearly.

Radical Expressions with Fractional Exponents

Choose the largest value:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:02 Start by picking the largest number you see.

00:06 Now, take the regular root and raise it to the power of two.

00:10 Next, combine everything into one root by multiplying the orders together.

00:20 Great! That's the value of the first expression.

00:24 Do the same steps for the next expressions to find out which is the biggest.

00:32 Well done! That's how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the largest value:

Step-by-step solution

To solve this problem, we'll follow the steps below:

Simplify each mathematical expression using exponent rules.
Compare the values derived from each simplification.

Let us analyze each given choice:

Choice 1: $\sqrt{\sqrt[6]{4}}$

The expression is simplified as follows: $(\sqrt[6]{4})^{\frac{1}{2}} = 4^{\frac{1}{6} \times \frac{1}{2}} = 4^{\frac{1}{12}}$ .

Choice 2: $\sqrt[6]{4}$

This expression is: $4^{\frac{1}{6}}$ .

Choice 3: $\sqrt[2]{\sqrt[3]{4}}$

Simplified, this is: $(\sqrt[3]{4})^{\frac{1}{2}} = 4^{\frac{1}{3} \times \frac{1}{2}} = 4^{\frac{1}{6}}$ .

Choice 4: $\sqrt{4}$

This expression is equivalent to: $4^{\frac{1}{2}}$ .

Now, let's compare the powers of 4:

Choice 1: $4^{\frac{1}{12}}$
Choice 2: $4^{\frac{1}{6}}$
Choice 3: $4^{\frac{1}{6}}$
Choice 4: $4^{\frac{1}{2}}$ - The largest exponent

The largest value among the given choices occurs when the exponent applied to the base 4 is maximized. Thus, the largest value is $\sqrt{4}$ .

Final Answer

$\sqrt{4}$

Key Points to Remember

Essential concepts to master this topic

Conversion Rule: Convert all radicals to fractional exponents for comparison
Technique: $\sqrt[6]{4} = 4^{\frac{1}{6}}$ and $\sqrt{\sqrt[6]{4}} = 4^{\frac{1}{12}}$
Check: Compare exponents with same base: $\frac{1}{2} > \frac{1}{6} > \frac{1}{12}$ ✓

Common Mistakes

Avoid these frequent errors

Comparing radical expressions without converting to same form
Don't try to compare $\sqrt{4}$ and $\sqrt[6]{4}$ directly = confusing comparison! Different radical forms make it hard to see which is larger. Always convert everything to fractional exponents with the same base first.

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 comparison much easier! When you have the same base (like 4), you just compare the exponents: $\frac{1}{2} > \frac{1}{6} > \frac{1}{12}$ .

How do I handle nested radicals like √(∛4)?

Work from inside out! First convert $\sqrt[3]{4} = 4^{\frac{1}{3}}$ , then $\sqrt{4^{\frac{1}{3}}} = (4^{\frac{1}{3}})^{\frac{1}{2}} = 4^{\frac{1}{6}}$ .

What's the rule for multiplying exponents in nested radicals?

When you have nested operations, multiply the exponents! For $\sqrt{\sqrt[6]{4}}$ , you get $4^{\frac{1}{6} \times \frac{1}{2}} = 4^{\frac{1}{12}}$ .

Why is √4 the largest value here?

Because $\sqrt{4} = 4^{\frac{1}{2}}$ has the largest exponent! Since $\frac{1}{2} = \frac{6}{12}$ is bigger than $\frac{1}{6} = \frac{2}{12}$ and $\frac{1}{12}$ .

Can I just calculate decimal values instead?

Yes, but it's less efficient! You'd need a calculator and might lose accuracy. The fractional exponent method is faster and shows the mathematical relationship clearly.

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