Mathematical Analysis: Finding the Largest Value in a Set

Q: Why do I need to simplify the fraction before taking the square root?

Simplifying first makes it much easier to see relationships! When you simplify $ \frac{4}{2} = 2 $, $ \frac{6}{3} = 2 $, and $ \frac{8}{4} = 2 $, you immediately see they're all equal.

Q: Can I just estimate which radical looks bigger?

Never guess! Radicals can be deceiving. $ \sqrt{\frac{8}{4}} $ might look largest because of the 8, but it equals the others. Always calculate to be sure.

Q: How do I know when all values are truly equal?

After simplifying, if every expression gives you the exact same result, then they're equal! In this problem, all three simplified to $ \sqrt{2} $.

Radical Simplification with Fraction Arguments

Choose the largest value

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Select the largest value

00:03 Calculate the quotient of each expression

00:07 Apply this method to each expression, and determine the largest one:

00:11 The larger the number in the root, the larger its value

00:14 This is 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 will simplify each square root expression separately and compare them:

Simplify $\sqrt{\frac{4}{2}}$ :

$\sqrt{\frac{4}{2}} = \sqrt{2}$

Simplify $\sqrt{\frac{6}{3}}$ :

$\sqrt{\frac{6}{3}} = \sqrt{2}$

Simplify $\sqrt{\frac{8}{4}}$ :

$\sqrt{\frac{8}{4}} = \sqrt{2}$

Upon simplification, each expression results in $\sqrt{2}$ . Therefore, all values are equal.

Thus, the correct choice is: All values are equal.

Final Answer

All values are equal

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify fractions inside square roots before taking the square root
Technique: $\sqrt{\frac{4}{2}} = \sqrt{2}$ by dividing 4÷2 first
Check: All expressions equal $\sqrt{2}$ when simplified correctly ✓

Common Mistakes

Avoid these frequent errors

Comparing radical expressions without simplifying
Don't compare $\sqrt{\frac{4}{2}}$ , $\sqrt{\frac{6}{3}}$ , and $\sqrt{\frac{8}{4}}$ in their original forms = wrong conclusions! The fractions look different but simplify to the same value. Always simplify the fraction inside the radical first, then evaluate.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that is equal to the following:

$\sqrt{a}:\sqrt{b}$

FAQ

Everything you need to know about this question

Why do I need to simplify the fraction before taking the square root?

Simplifying first makes it much easier to see relationships! When you simplify $\frac{4}{2} = 2$ , $\frac{6}{3} = 2$ , and $\frac{8}{4} = 2$ , you immediately see they're all equal.

Can I just estimate which radical looks bigger?

Never guess! Radicals can be deceiving. $\sqrt{\frac{8}{4}}$ might look largest because of the 8, but it equals the others. Always calculate to be sure.

What if the fraction doesn't simplify to a whole number?

That's fine! You'd still compare the simplified radicals. For example, $\sqrt{\frac{5}{2}}$ and $\sqrt{\frac{10}{4}}$ both equal $\sqrt{2.5}$ .

How do I know when all values are truly equal?

After simplifying, if every expression gives you the exact same result, then they're equal! In this problem, all three simplified to $\sqrt{2}$ .

Should I use a calculator for these comparisons?

Try simplifying by hand first! It builds your skills. You can use a calculator to verify that $\sqrt{2} ≈ 1.414$ for all three expressions.

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