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:

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

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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

Choose the expression that is equal to the following:

$\sqrt{a}:\sqrt{b}$

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 $36x^2$ .

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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

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

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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?