Simplify the Radical Expression: (√10 × √30)/√100

Q: Why can I multiply the numbers inside the square roots?

The multiplication property of radicals states that $ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b} $ when both numbers are positive. This lets you combine radicals into a single expression!

Q: How do I know when to use the division property?

Use the division property $ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $ when you have radicals in both numerator and denominator. It simplifies the fraction into one radical expression.

Q: What if I get a perfect square under the radical?

Perfect squares simplify completely! For example, $ \sqrt{100} = 10 $ because $ 10^2 = 100 $. Always check if your final radical can be simplified further.

Q: Should I always combine radicals first?

Yes! When multiplying or dividing radicals, combine them first using the properties. This usually makes the problem easier to solve than working with multiple separate radicals.

Radical Expressions with Multiplication and Division Properties

Solve the following exercise:

$\frac{\sqrt{10}\cdot\sqrt{30}}{\sqrt{100}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:08 When multiplying the square root of a number (A) by the square root of another number (B)

00:12 The result equals the square root of their product (A times B)

00:17 Breakdown 100 into factors of 10 and 10

00:22 Apply this formula to our exercise

00:26 Simplify wherever possible

00:34 The square root of the numerator (A) divided by the square root of the denominator (B)

00:39 Equals the square root of the entire fraction (A divided by B)

00:43 Apply this formula to our exercise

00:49 Calculate 30 divided by 10

00:52 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:

$\frac{\sqrt{10}\cdot\sqrt{30}}{\sqrt{100}}=$

Step-by-step solution

To solve the problem $\frac{\sqrt{10} \cdot \sqrt{30}}{\sqrt{100}}$ , we'll use the rules of square roots, specifically the multiplication and division properties.

Start by simplifying the numerator using the multiplication property of square roots:

$\sqrt{10} \cdot \sqrt{30} = \sqrt{10 \times 30} = \sqrt{300}$

Next, we simplify the entire fraction using the division property of square roots:

$\frac{\sqrt{300}}{\sqrt{100}} = \sqrt{\frac{300}{100}} = \sqrt{3}$

Thus, the simplified form of the expression $\frac{\sqrt{10} \cdot \sqrt{30}}{\sqrt{100}}$ is $\sqrt{3}$ .

Therefore, the solution to the problem is $\sqrt{3}$ .

Final Answer

$\sqrt{3}$

Key Points to Remember

Essential concepts to master this topic

Multiplication Rule: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b}$ for positive values
Technique: Combine radicals first: $\sqrt{10} \cdot \sqrt{30} = \sqrt{300}$
Check: Verify $\sqrt{3} \cdot \sqrt{3} = 3$ and $\sqrt{100} = 10$ ✓

Common Mistakes

Avoid these frequent errors

Simplifying radicals before applying properties
Don't break down $\sqrt{10}$ and $\sqrt{30}$ separately first = makes problem unnecessarily complex! This creates extra steps and increases chance for errors. Always use the multiplication property first: $\sqrt{10} \cdot \sqrt{30} = \sqrt{300}$ , then simplify.

Practice Quiz

Test your knowledge with interactive questions

Choose the largest value

FAQ

Everything you need to know about this question

Why can I multiply the numbers inside the square roots?

The multiplication property of radicals states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b}$ when both numbers are positive. This lets you combine radicals into a single expression!

How do I know when to use the division property?

Use the division property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ when you have radicals in both numerator and denominator. It simplifies the fraction into one radical expression.

What if I get a perfect square under the radical?

Perfect squares simplify completely! For example, $\sqrt{100} = 10$ because $10^2 = 100$ . Always check if your final radical can be simplified further.

Should I always combine radicals first?

Yes! When multiplying or dividing radicals, combine them first using the properties. This usually makes the problem easier to solve than working with multiple separate radicals.

How can I check if my answer is correct?

Square your final answer and see if it matches the simplified fraction. Here: $(\sqrt{3})^2 = 3$ and $\frac{300}{100} = 3$ ✓

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Simplify the Radical Expression: (√10 × √30)/√100

Radical Expressions with Multiplication and Division Properties

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can I multiply the numbers inside the square roots?

How do I know when to use the division property?

What if I get a perfect square under the radical?

Should I always combine radicals first?

How can I check if my answer is correct?

🌟 Unlock Your Math Potential

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