Solve for X: Simplifying √20·√5/x = 2√25 Equation

Q: Can I multiply the square roots directly like regular numbers?

Yes, but with the product rule! When multiplying square roots, use $ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} $. So $ \sqrt{20} \cdot \sqrt{5} = \sqrt{20 \times 5} = \sqrt{100} = 10 $.

Q: How do I know when to use the product rule for square roots?

Use the product rule whenever you see two or more square roots being multiplied. This lets you combine them into one radical, making calculations much easier!

Q: Why did we get x = 1 as the answer?

After simplifying both sides to 10, we had $ \frac{10}{x} = 10 $. Cross-multiplying gives us $ 10 = 10x $, so $ x = 1 $. Always verify by substituting back!

Q: What's the difference between √25 and 2√25?

$ \sqrt{25} = 5 $ (just the square root), but $ 2\sqrt{25} = 2 \times 5 = 10 $ (multiply by 2). The coefficient multiplies the entire radical value.

Radical Equations with Product Rule Simplification

Solve for x:

$\frac{\sqrt{20}\cdot\sqrt{5}}{x}=2\cdot\sqrt{25}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the value X

00:03 Isolate X

00:28 When multiplying a square root of a number (A) by a square root of another number (B)

00:32 The result equals the square root of their product (A times B)

00:37 Apply this formula to our exercise and calculate the product

00:44 Break down 100 into 10 squared

00:48 Break down 25 into 5 squared

00:51 Every square root of a number (A) squared equals the number itself (A)

00:54 Apply this formula to our exercise and cancel out the squares

01:05 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

$\frac{\sqrt{20}\cdot\sqrt{5}}{x}=2\cdot\sqrt{25}$

Step-by-step solution

To solve the equation \< $\frac{\sqrt{20}\cdot\sqrt{5}}{x} = 2\cdot\sqrt{25}$ \>, follow these steps:

Step 1: Simplify the left-hand side.
- Use the product rule for roots: $\sqrt{20} \cdot \sqrt{5} = \sqrt{20 \cdot 5} = \sqrt{100}$ .
Step 2: Simplify $\sqrt{100}$ to get $10\.$
Step 3: Substitute and simplify the equation:
$\(\frac{10}{x} = 2 \cdot \sqrt{25}$ .
Step 4: Simplify the right-hand side:
$\sqrt{25} = 5$ so $2 \cdot 5 = 10$ .
Step 5: Equate both sides:
$\frac{10}{x} = 10$ .
Step 6: Solve for $x$ :
Multiply both sides by $x$ , then divide by 10:
$10 = 10x$ produces $x = 1$ .

Therefore, the solution to the problem is $\boldsymbol{x = 1}$ .

Final Answer

$1$

Key Points to Remember

Essential concepts to master this topic

Product Rule: Combine radicals: $\sqrt{20} \cdot \sqrt{5} = \sqrt{100} = 10$
Technique: Simplify both sides first: $2\sqrt{25} = 2 \cdot 5 = 10$
Check: Substitute x = 1 back: $\frac{10}{1} = 10$ matches $2\sqrt{25} = 10$ ✓

Common Mistakes

Avoid these frequent errors

Solving without simplifying radicals first
Don't leave $\sqrt{20} \cdot \sqrt{5}$ unsimplified = confusing work with wrong setup! This makes the equation unnecessarily complex and increases calculation errors. Always use the product rule to simplify $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ before solving.

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

Can I multiply the square roots directly like regular numbers?

Yes, but with the product rule! When multiplying square roots, use $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ . So $\sqrt{20} \cdot \sqrt{5} = \sqrt{20 \times 5} = \sqrt{100} = 10$ .

What if I can't simplify the square root to a whole number?

That's okay! Not all square roots simplify to whole numbers. Just make sure to combine them correctly using the product rule, then work with the simplified form throughout your solution.

How do I know when to use the product rule for square roots?

Use the product rule whenever you see two or more square roots being multiplied. This lets you combine them into one radical, making calculations much easier!

Why did we get x = 1 as the answer?

After simplifying both sides to 10, we had $\frac{10}{x} = 10$ . Cross-multiplying gives us $10 = 10x$ , so $x = 1$ . Always verify by substituting back!

What's the difference between √25 and 2√25?

$\sqrt{25} = 5$ (just the square root), but $2\sqrt{25} = 2 \times 5 = 10$ (multiply by 2). The coefficient multiplies the entire radical value.

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Solve for X: Simplifying √20·√5/x = 2√25 Equation

Radical Equations with Product Rule Simplification

❤️ 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

Can I multiply the square roots directly like regular numbers?

What if I can't simplify the square root to a whole number?

How do I know when to use the product rule for square roots?

Why did we get x = 1 as the answer?

What's the difference between √25 and 2√25?

🌟 Unlock Your Math Potential

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