Solve for X: Simplifying √8·√4·√2/√2 = √(x²)

Q: Why can I combine all the square roots under one radical?

The product rule for radicals says $ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} $. This works because both expressions represent the same value - the positive number that when squared gives you the product inside.

Q: How do I divide square roots like $ \frac{\sqrt{64}}{\sqrt{2}} $ ?

Use the quotient rule: $ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $. So $ \frac{\sqrt{64}}{\sqrt{2}} = \sqrt{\frac{64}{2}} = \sqrt{32} $.

Q: Why does $ \sqrt{x^2} $ equal $ |x| $ and not just x?

Because $ \sqrt{x^2} $ always gives the positive square root. For example, if x = -5, then $ \sqrt{(-5)^2} = \sqrt{25} = 5 $, not -5. That's why we write $ |x| $ to include both positive and negative solutions.

Q: Should I leave my answer as $ \sqrt{32} $ or simplify it further?

You can simplify further! $ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} $. However, since the answer choices show $ \sqrt{32} $, either form is correct.

Q: What if I calculated $ 8 \cdot 4 \cdot 2 $ wrong?

Double-check your multiplication step by step: $ 8 \times 4 = 32 $, then $ 32 \times 2 = 64 $. Getting the wrong product here will mess up your entire answer!

Radical Simplification with Product Rules

Solve for x:

$\frac{\sqrt{8}\cdot\sqrt{4}\cdot\sqrt{2}}{\sqrt{2}}=\sqrt{x^2}$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the value X

00:03 Simplify wherever possible

00:11 When multiplying the root of a number (A) by the root of another number (B)

00:14 The result equals the root of their product (A times B)

00:18 Apply this formula to our exercise

00:26 Any root of a number (A) squared equals the number itself (A)

00:30 Apply this formula to our exercise, and cancel out the square:

00:34 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

$\frac{\sqrt{8}\cdot\sqrt{4}\cdot\sqrt{2}}{\sqrt{2}}=\sqrt{x^2}$

Step-by-step solution

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

Simplify the expression $\sqrt{8}\cdot\sqrt{4}\cdot\sqrt{2}$ .
Divide this product by $\sqrt{2}$ .
Set the result equal to $\sqrt{x^2}$ and solve for $x$ .

Now, let's work through each step:

Step 1: First, simplify the product under the square root:

$\sqrt{8}\cdot\sqrt{4}\cdot\sqrt{2} = \sqrt{8\cdot4\cdot2}$ .

This simplifies to:

$\sqrt{64}$ , because $8 \cdot 4 \cdot 2 = 64$ .

Step 2: Now, divide by $\sqrt{2}$ :

$\frac{\sqrt{64}}{\sqrt{2}} = \sqrt{\frac{64}{2}} = \sqrt{32}$ .

Step 3: Equate this to $\sqrt{x^2}$ :

$\sqrt{32} = \sqrt{x^2}$ .

This implies $|x| = \sqrt{32}$ , giving us two possible solutions: $x = \sqrt{32}$ and $x = -\sqrt{32}$ .

Since the simplification naturally leads to positive expressions, we find:

The solution is $x = \sqrt{32}$ .

Final Answer

$x=\sqrt{32}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: Multiply radicals by combining under one radical sign
Technique: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ , so $\sqrt{8} \cdot \sqrt{4} \cdot \sqrt{2} = \sqrt{64}$
Check: Verify $\sqrt{32} = \sqrt{x^2}$ means $|x| = \sqrt{32}$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying under the radical
Don't write $\sqrt{8} \cdot \sqrt{4} \cdot \sqrt{2} = \sqrt{8+4+2} = \sqrt{14}$ ! This gives the wrong answer because you're adding instead of multiplying. Always use the product rule: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ .

Practice Quiz

Test your knowledge with interactive questions

Choose the largest value

FAQ

Everything you need to know about this question

Why can I combine all the square roots under one radical?

The product rule for radicals says $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ . This works because both expressions represent the same value - the positive number that when squared gives you the product inside.

How do I divide square roots like $\frac{\sqrt{64}}{\sqrt{2}}$ ?

Use the quotient rule: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ . So $\frac{\sqrt{64}}{\sqrt{2}} = \sqrt{\frac{64}{2}} = \sqrt{32}$ .

Why does $\sqrt{x^2}$ equal $|x|$ and not just x?

Because $\sqrt{x^2}$ always gives the positive square root. For example, if x = -5, then $\sqrt{(-5)^2} = \sqrt{25} = 5$ , not -5. That's why we write $|x|$ to include both positive and negative solutions.

Should I leave my answer as $\sqrt{32}$ or simplify it further?

You can simplify further! $\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$ . However, since the answer choices show $\sqrt{32}$ , either form is correct.

What if I calculated $8 \cdot 4 \cdot 2$ wrong?

Double-check your multiplication step by step: $8 \times 4 = 32$ , then $32 \times 2 = 64$ . Getting the wrong product here will mess up your entire answer!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rules of Roots questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Solve for X: Simplifying √8·√4·√2/√2 = √(x²)

Radical Simplification with Product Rules

❤️ 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 combine all the square roots under one radical?

How do I divide square roots like $\frac{\sqrt{64}}{\sqrt{2}}$ ?

Why does $\sqrt{x^2}$ equal $|x|$ and not just x?

Should I leave my answer as $\sqrt{32}$ or simplify it further?

What if I calculated $8 \cdot 4 \cdot 2$ wrong?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

Solve for X: Simplifying √4·√5/√25 Equation Step-by-Step

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

Simplify the Square Root Expression: √(12x⁴)

Solve the Square Root Expression: Simplifying √(25x⁴)

Square Root Quotient Property

Solve for X: Square Root Equation sqrt(2)·sqrt(4)/sqrt(16) = x/sqrt(8)

Solve the Square Root Equation: Finding X in √6x = √36

Solve the Square Root Expression: Simplifying √(2/4)

Solve Square Root: Simplifying √(225/25) Step by Step

Solve for X: Simplifying √8·√4·√2/√2 = √(x²)

Radical Simplification with Product Rules

❤️ 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 combine all the square roots under one radical?

How do I divide square roots like 642 \frac{\sqrt{64}}{\sqrt{2}} 2​64​​?

Why does x2 \sqrt{x^2} x2​ equal ∣x∣ |x| ∣x∣ and not just x?

Should I leave my answer as 32 \sqrt{32} 32​ or simplify it further?

What if I calculated 8⋅4⋅2 8 \cdot 4 \cdot 2 8⋅4⋅2 wrong?

🌟 Unlock Your Math Potential

More Questions

Rules of Roots Combined

Solve for X: Simplifying √4·√5/√25 Equation Step-by-Step

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

Simplify the Square Root Expression: √(12x⁴)

Solve the Square Root Expression: Simplifying √(25x⁴)

Square Root Quotient Property

Solve for X: Square Root Equation sqrt(2)·sqrt(4)/sqrt(16) = x/sqrt(8)

Solve the Square Root Equation: Finding X in √6x = √36

Solve the Square Root Expression: Simplifying √(2/4)

Solve Square Root: Simplifying √(225/25) Step by Step

How do I divide square roots like $\frac{\sqrt{64}}{\sqrt{2}}$ ?

Why does $\sqrt{x^2}$ equal $|x|$ and not just x?

Should I leave my answer as $\sqrt{32}$ or simplify it further?

What if I calculated $8 \cdot 4 \cdot 2$ wrong?