Simplify sqrt(36x⁴)/sqrt(x²): Square Root Fraction Problem

Q: Why does $ \sqrt{x^4} = x^2 $ and not $ x^4 $ ?

The square root halves the exponent! Since $ \sqrt{x^4} = (x^4)^{1/2} = x^{4/2} = x^2 $. Think of it as: what number squared gives $ x^4 $? That's $ x^2 $!

Q: Can I use the property $ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $ ?

Yes! You could write $ \frac{\sqrt{36x^4}}{\sqrt{x^2}} = \sqrt{\frac{36x^4}{x^2}} = \sqrt{36x^2} = 6x $. This gives the same answer but requires careful fraction simplification inside the square root.

Q: What if the variable has a negative value?

For this problem, we assume $ x \geq 0 $ to keep things simple. When dealing with negative values, $ \sqrt{x^2} = |x| $ (absolute value), which makes the problem more complex.

Q: How do I remember $ \sqrt{36} = 6 $ ?

Practice your perfect squares! Remember: $ 6^2 = 36 $, so $ \sqrt{36} = 6 $. Other key ones: $ \sqrt{25} = 5 $, $ \sqrt{49} = 7 $, $ \sqrt{64} = 8 $.

Q: Why don't I get $ 36x^2 $ as the final answer?

That would be $ \sqrt{36x^4} $ without the division! Don't forget to divide by $ \sqrt{x^2} = x $. So $ \frac{6x^2}{x} = 6x $, not $ 36x^2 $.

Square Root Simplification with Variable Exponents

Solve the following exercise:

$\frac{\sqrt{36x^4}}{\sqrt{x^2}}=$

❤️ 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

00:09 Let's simplify this problem together.

00:12 When taking the square root of multiplied terms, like A times B,

00:17 we can find the root of A, then multiply it by the root of B.

00:22 Now, let's apply this to our exercise.

00:29 We can rewrite 36 as 6 squared.

00:34 And X to the 4th power as X squared, squared.

00:40 Remember, the root of A squared removes the square.

00:45 Let's apply this and remove the squares in our problem.

01:00 Rewrite X squared as X times X.

01:09 Simplify wherever you can.

01:12 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\frac{\sqrt{36x^4}}{\sqrt{x^2}}=$

Step-by-step solution

To solve the problem, we will simplify the expression step-by-step:

Step 1: Simplify $\sqrt{36x^4}$
We know that $\sqrt{36x^4}$ can be rewritten as $\sqrt{36} \cdot \sqrt{x^4}$ .
First, simplify $\sqrt{36}$ :
$\sqrt{36} = 6$ because $6^2 = 36$ .
Next, simplify $\sqrt{x^4}$ :
$\sqrt{x^4} = x^2$ because $(x^2)^2 = x^4$ .
Step 2: Simplify $\sqrt{x^2}$
$\sqrt{x^2} = x$ (assuming $x \geq 0$ to avoid absolute values).
Step 3: Simplify the expression $\frac{\sqrt{36x^4}}{\sqrt{x^2}}$
Substitute the values obtained in the above steps:
$\frac{\sqrt{36x^4}}{\sqrt{x^2}} = \frac{6x^2}{x}$ .
The expression simplifies to $6x$ as $\frac{x^2}{x} = x$ .

Therefore, the solution to the problem is $6x$ , which matches choice 4.

Final Answer

$6x$

Key Points to Remember

Essential concepts to master this topic

Rule: $\sqrt{a^n} = a^{n/2}$ when simplifying square roots with exponents
Technique: $\sqrt{36x^4} = 6x^2$ and $\sqrt{x^2} = x$ before dividing
Check: Substitute $x = 2$ : $\frac{\sqrt{36 \cdot 16}}{\sqrt{4}} = \frac{24}{2} = 12 = 6(2)$ ✓

Common Mistakes

Avoid these frequent errors

Dividing exponents instead of simplifying square roots first
Don't divide $x^4$ by $x^2$ to get $x^2$ inside the square root = wrong answer $6x$ ! This ignores the square root operations. Always simplify each square root separately first, then divide the results.

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 does $\sqrt{x^4} = x^2$ and not $x^4$ ?

The square root halves the exponent! Since $\sqrt{x^4} = (x^4)^{1/2} = x^{4/2} = x^2$ . Think of it as: what number squared gives $x^4$ ? That's $x^2$ !

Can I use the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ ?

Yes! You could write $\frac{\sqrt{36x^4}}{\sqrt{x^2}} = \sqrt{\frac{36x^4}{x^2}} = \sqrt{36x^2} = 6x$ . This gives the same answer but requires careful fraction simplification inside the square root.

What if the variable has a negative value?

For this problem, we assume $x \geq 0$ to keep things simple. When dealing with negative values, $\sqrt{x^2} = |x|$ (absolute value), which makes the problem more complex.

How do I remember $\sqrt{36} = 6$ ?

Practice your perfect squares! Remember: $6^2 = 36$ , so $\sqrt{36} = 6$ . Other key ones: $\sqrt{25} = 5$ , $\sqrt{49} = 7$ , $\sqrt{64} = 8$ .

Why don't I get $36x^2$ as the final answer?

That would be $\sqrt{36x^4}$ without the division! Don't forget to divide by $\sqrt{x^2} = x$ . So $\frac{6x^2}{x} = 6x$ , not $36x^2$ .

🌟 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

Simplify sqrt(36x⁴)/sqrt(x²): Square Root Fraction Problem

Square Root Simplification with Variable Exponents

❤️ 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 does $\sqrt{x^4} = x^2$ and not $x^4$ ?

Can I use the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ ?

What if the variable has a negative value?

How do I remember $\sqrt{36} = 6$ ?

Why don't I get $36x^2$ as the final answer?

🌟 Unlock Your Math Potential

More Questions

Square Root Quotient Property

Simplify the Radical Expression: √(49x²)/x

Simplify sqrt(25x²)/sqrt(x²): Step-by-Step Radical Reduction

Simplify the Square Root: √(100x⁴/25x²) Step-by-Step Solution

Simplify the Expression: √(x⁴)/x Step-by-Step Solution

Simplify sqrt(36x⁴)/sqrt(x²): Square Root Fraction Problem

Square Root Simplification with Variable Exponents

❤️ 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 does x4=x2 \sqrt{x^4} = x^2 x4​=x2 and not x4 x^4 x4?

Can I use the property ab=ab \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} b​a​​=ba​​?

What if the variable has a negative value?

How do I remember 36=6 \sqrt{36} = 6 36​=6?

Why don't I get 36x2 36x^2 36x2 as the final answer?

🌟 Unlock Your Math Potential

More Questions

Square Root Quotient Property

Simplify the Radical Expression: √(49x²)/x

Simplify sqrt(25x²)/sqrt(x²): Step-by-Step Radical Reduction

Simplify the Square Root: √(100x⁴/25x²) Step-by-Step Solution

Simplify the Expression: √(x⁴)/x Step-by-Step Solution

Why does $\sqrt{x^4} = x^2$ and not $x^4$ ?

Can I use the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ ?

How do I remember $\sqrt{36} = 6$ ?

Why don't I get $36x^2$ as the final answer?