Simplify Square Root Expression: √(144x¹⁰/9x⁴)

Q: Why do I need to simplify inside the radical first?

Simplifying before taking the square root makes the problem much easier! $ \sqrt{\frac{144x^{10}}{9x^4}} = \sqrt{16x^6} $ is simpler than trying to handle $ \frac{\sqrt{144x^{10}}}{\sqrt{9x^4}} $ separately.

Q: How do I handle the variable exponents in square roots?

Remember that $ \sqrt{x^n} = x^{n/2} $. So $ \sqrt{x^6} = x^{6/2} = x^3 $. The square root halves the exponent!

Q: What if the exponent under the square root is odd?

If you have an odd exponent like $ x^5 $, you can write it as $ x^4 \cdot x = (x^2)^2 \cdot x $. Then $ \sqrt{x^5} = x^2\sqrt{x} $.

Q: Can I just divide the numbers and variables separately?

Yes! This is exactly the right approach. Divide coefficients: $ \frac{144}{9} = 16 $, and subtract exponents: $ \frac{x^{10}}{x^4} = x^{10-4} = x^6 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this math problem together.

00:14 When we have the square root of a fraction, A divided by B,

00:18 we can rewrite it as the square root of A divided by the square root of B.

00:23 Now, we'll apply this method to our exercise. Let's continue!

00:29 For the square root of a multiplication, A times B,

00:34 split it as the square root of A times the square root of B.

00:39 Let's use this formula in our exercise and move forward.

00:48 First, break down one hundred forty-four into twelve squared.

00:57 Now, break down nine into three squared.

01:04 Remember, the square root of any A squared is just A.

01:09 Apply this to our problem and cancel out the squares.

01:21 Let's break X to the power of ten into X to the power of five squared.

01:27 And break X to the power of four into X squared squared.

01:34 Simplify the squares using the roots, step by step.

01:43 Break down twelve into the factors, four and three.

01:49 Break X to the power of five into X cubed times X squared.

01:55 Simplify everything you can. Great work!

01:59 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve the following exercise:

$\sqrt{\frac{144x^{10}}{9x^4}}=$

2

Step-by-step solution

To solve this problem, we will proceed as follows:

Step 1: Simplify the expression inside the square root.
Step 2: Apply the square root to both the numerator and the denominator separately.
Step 3: Simplify the resulting expression.

Let's go through each step:

Step 1: Simplify the fraction $\frac{144x^{10}}{9x^4}$ .

In the given expression $\frac{144x^{10}}{9x^4}$ , separate the numeric part from the variable part:

$\frac{144}{9} = 16$ and $\frac{x^{10}}{x^4} = x^{10-4} = x^6$ .

Thus, the expression simplifies to $16x^6$ .

Step 2: Apply the square root property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to each part.

$\sqrt{\frac{144x^{10}}{9x^4}} = \sqrt{16x^6}$ .

Now separate into numeric and variable components: $\sqrt{16}\cdot\sqrt{x^6}$ .

$\sqrt{16} = 4$ because $4^2 = 16$ .

$\sqrt{x^6} = (x^6)^{1/2} = x^{6/2} = x^3$ .

Step 3: Combine the results.

Combine the simplified components: $4x^3$ .

Therefore, the solution to the problem is $\boxed{4x^3}$ .

3

Final Answer

$4x^3$

Key Points to Remember

Essential concepts to master this topic

Fraction Rule: Simplify inside the radical before taking square root
Technique: Divide coefficients (144÷9=16) and subtract exponents (x¹⁰÷x⁴=x⁶)
Check: Verify (4x³)² = 16x⁶ equals simplified expression inside radical ✓

Common Mistakes

Avoid these frequent errors

Taking square root of numerator and denominator separately without simplifying first
Don't calculate √144x¹⁰ ÷ √9x⁴ = 12x⁵ ÷ 3x² and get messy fractions! This creates unnecessary complexity and often leads to calculation errors. Always simplify the fraction inside the radical first to get √(16x⁶) = 4x³.

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 do I need to simplify inside the radical first?

+

Simplifying before taking the square root makes the problem much easier! $\sqrt{\frac{144x^{10}}{9x^4}} = \sqrt{16x^6}$ is simpler than trying to handle $\frac{\sqrt{144x^{10}}}{\sqrt{9x^4}}$ separately.

How do I handle the variable exponents in square roots?

+

Remember that $\sqrt{x^n} = x^{n/2}$ . So $\sqrt{x^6} = x^{6/2} = x^3$ . The square root halves the exponent!

What if the exponent under the square root is odd?

+

If you have an odd exponent like $x^5$ , you can write it as $x^4 \cdot x = (x^2)^2 \cdot x$ . Then $\sqrt{x^5} = x^2\sqrt{x}$ .

Can I just divide the numbers and variables separately?

+

Yes! This is exactly the right approach. Divide coefficients: $\frac{144}{9} = 16$ , and subtract exponents: $\frac{x^{10}}{x^4} = x^{10-4} = x^6$ .

How do I check my answer is correct?

+

Square your final answer and see if it equals the simplified expression inside the original radical. $(4x^3)^2 = 16x^6$ ✓ matches our simplified fraction!

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Simplify Square Root Expression: √(144x¹⁰/9x⁴)

Radical Simplification with Variable Exponents

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