Number Comparison: Determining the Largest Value in a Set

Q: Why does a smaller radical index give a larger value?

Think of it this way: $ \sqrt{2} $ asks "what number squared gives 2?" while $ \sqrt[4]{2} $ asks "what number to the 4th power gives 2?" The 4th power grows faster, so you need a smaller base number!

Q: How do I convert radicals to exponential form?

Use the rule: $ \sqrt[n]{a} = a^{1/n} $. So $ \sqrt{2} = 2^{1/2} $, $ \sqrt[3]{2} = 2^{1/3} $, and so on. The index becomes the denominator of the exponent!

Q: Can I just use a calculator to compare these?

Yes, but understanding the exponential method helps you solve similar problems without a calculator! $ \sqrt{2} ≈ 1.414 $, $ \sqrt[3]{2} ≈ 1.26 $, etc.

Q: What if the bases were different numbers?

If comparing $ \sqrt{3} $ vs $ \sqrt[3]{4} $, you'd need to calculate the actual values since the bases are different. The exponential method works best when bases are the same!

Q: Is there a pattern for which radical is largest?

Yes! For the same positive base greater than 1, $ \sqrt[2]{a} > \sqrt[3]{a} > \sqrt[4]{a} > \sqrt[5]{a} $... The smaller the index, the larger the value!

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 Let's start by choosing the largest value.

00:06 A regular root raised to the power of two, means square it.

00:11 Think of a square root, it's like taking a number to the power of one half.

00:17 Now, use this method on the next expressions to find the largest one.

00:22 And there you have it, that's the solution.

00:25 Chapter title.

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 these steps:

Step 1: Convert each expression into exponential form.
Step 2: Compare the resulting expressions by examining the exponents.
Step 3: Identify which expression corresponds to the largest value.

Let's work through the solution:

Step 1: Convert each root to exponential form:
- $\sqrt{2} = 2^{0.5}$
- $\sqrt[3]{2} = 2^{1/3} \approx 2^{0.333}$
- $\sqrt[4]{2} = 2^{1/4} \approx 2^{0.25}$
- $\sqrt[5]{2} = 2^{1/5} \approx 2^{0.2}$

Step 2: Compare the exponents $0.5$ , $0.333$ , $0.25$ , and $0.2$ . Clearly, $0.5$ is the largest among these values.

Step 3: The expression with the largest exponent is $\sqrt{2} = 2^{0.5}$ , so $\sqrt{2}$ is the largest value.

Therefore, the solution to the problem is $\sqrt{2}$ .

Final Answer

$\sqrt{2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert radicals to exponential form for easy comparison
Technique: $\sqrt[n]{a} = a^{1/n}$ , so $\sqrt{2} = 2^{1/2} = 2^{0.5}$
Check: Compare exponents: 0.5 > 0.333 > 0.25 > 0.2, so $\sqrt{2}$ is largest ✓

Common Mistakes

Avoid these frequent errors

Comparing radical expressions directly without converting to exponential form
Don't try to compare $\sqrt{2}$ vs $\sqrt[3]{2}$ by guessing = wrong conclusions! Different radical indices make direct comparison impossible. Always convert to exponential form first: $2^{1/2}$ vs $2^{1/3}$ , then compare the exponents.

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 does a smaller radical index give a larger value?

Think of it this way: $\sqrt{2}$ asks "what number squared gives 2?" while $\sqrt[4]{2}$ asks "what number to the 4th power gives 2?" The 4th power grows faster, so you need a smaller base number!

How do I convert radicals to exponential form?

Use the rule: $\sqrt[n]{a} = a^{1/n}$ . So $\sqrt{2} = 2^{1/2}$ , $\sqrt[3]{2} = 2^{1/3}$ , and so on. The index becomes the denominator of the exponent!

Can I just use a calculator to compare these?

Yes, but understanding the exponential method helps you solve similar problems without a calculator! $\sqrt{2} ≈ 1.414$ , $\sqrt[3]{2} ≈ 1.26$ , etc.

What if the bases were different numbers?

If comparing $\sqrt{3}$ vs $\sqrt[3]{4}$ , you'd need to calculate the actual values since the bases are different. The exponential method works best when bases are the same!

Is there a pattern for which radical is largest?

Yes! For the same positive base greater than 1, $\sqrt[2]{a} > \sqrt[3]{a} > \sqrt[4]{a} > \sqrt[5]{a}$ ... The smaller the index, the larger the value!

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