Simplify the Radical Expression: √10/√2 Step-by-Step

Q: When can I use the quotient property for square roots?

You can use $ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $ whenever both numbers under the square roots are positive. This makes complex radical divisions much simpler!

Q: How do I know if √5 is simplified?

Check if 5 has any perfect square factors. Since 5 = 1 × 5 and neither 1 nor 5 are perfect squares (except 1), $ \sqrt{5} $ is fully simplified.

Q: Could I rationalize the denominator instead?

Yes! You could multiply by $ \frac{\sqrt{2}}{\sqrt{2}} $ to get $ \frac{\sqrt{20}}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5} $. Both methods give the same answer!

Q: Why is the answer √5 and not a decimal?

In algebra, we keep answers in exact form when possible. $ \sqrt{5} $ is exact, while 2.236... is just an approximation. Always give exact answers unless told to use decimals.

Radical Quotient Property with Square Roots

Solve the following exercise:

$\frac{\sqrt{10}}{\sqrt{2}}=$

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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 Break down 10 into factors of 5 and 2

00:09 When we have a root of the multiplied terms (A times B)

00:13 We can convert it to the multiplication of the root(A²) times the root(B)

00:17 Apply this formula to our exercise and proceed to convert to 2 roots:

00:26 Simplify wherever possible

00:29 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:

$\frac{\sqrt{10}}{\sqrt{2}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Apply the square root quotient property.
Step 2: Simplify the fraction under the square root.
Step 3: Evaluate the square root if possible.

Now, let's work through each step:
Step 1: The square root quotient property tells us that $\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}}$ .
Step 2: Simplify the fraction inside the square root: $\frac{10}{2} = 5$ .
Step 3: Therefore, $\sqrt{\frac{10}{2}} = \sqrt{5}$ .

Therefore, the solution to the problem is $\sqrt{5}$ .

Final Answer

$\sqrt{5}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ for positive values
Technique: Simplify fraction inside: $\frac{10}{2} = 5$ , then evaluate
Check: Verify $(\sqrt{5})^2 = 5$ and $\frac{10}{2} = 5$ match ✓

Common Mistakes

Avoid these frequent errors

Dividing the numbers under the square roots separately
Don't calculate $\sqrt{10} ÷ \sqrt{2}$ as decimal approximations = messy, wrong answers! This creates unnecessary complexity and rounding errors. Always use the quotient property first: $\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

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

FAQ

Everything you need to know about this question

Why can't I just divide 10 by 2 to get 5?

You're on the right track! You do divide 10 by 2 to get 5, but it stays under the square root. So $\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{5}$ , not just 5.

When can I use the quotient property for square roots?

You can use $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ whenever both numbers under the square roots are positive. This makes complex radical divisions much simpler!

How do I know if √5 is simplified?

Check if 5 has any perfect square factors. Since 5 = 1 × 5 and neither 1 nor 5 are perfect squares (except 1), $\sqrt{5}$ is fully simplified.

Could I rationalize the denominator instead?

Yes! You could multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ to get $\frac{\sqrt{20}}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5}$ . Both methods give the same answer!

Why is the answer √5 and not a decimal?

In algebra, we keep answers in exact form when possible. $\sqrt{5}$ is exact, while 2.236... is just an approximation. Always give exact answers unless told to use decimals.

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