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

Q: Why do I need to simplify the fraction before taking the square root?

Simplifying first makes the problem much easier! $ \sqrt{4x^2} $ is simpler to work with than $ \sqrt{\frac{100x^4}{25x^2}} $. Plus, you're less likely to make calculation errors.

Q: How do I divide variables with exponents like x⁴/x²?

Use the quotient rule for exponents: $ \frac{x^a}{x^b} = x^{a-b} $. So $ \frac{x^4}{x^2} = x^{4-2} = x^2 $. Just subtract the bottom exponent from the top!

Q: What if x is negative? Does √(x²) still equal x?

Technically, $ \sqrt{x^2} = |x| $ (absolute value of x). But in most algebra problems, we assume variables are positive unless stated otherwise, so $ \sqrt{x^2} = x $ works fine.

Q: Can I cancel terms directly in the original expression?

Yes! You can cancel $ x^2 $ from top and bottom: $ \frac{100x^4}{25x^2} = \frac{100x^2}{25} = 4x^2 $. This gives the same result as using exponent rules.

Q: How do I check if 2x is the right answer?

Square your answer: $ (2x)^2 = 4x^2 $. This should match what you got after simplifying the fraction under the square root. If they're equal, you're correct!

Square Root Simplification with Rational Expressions

Solve the following exercise:

$\sqrt{\frac{100x^4}{25x^2}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When there's a root of the multiplied terms (A times B)

00:06 We can break it down to root(A²²²) X root(B)

00:09 We'll apply this formula to our exercise

00:24 Break down 100 into 10 squared

00:27 Break down X⁴ into (X²)²

00:34 reak down 25 into 5 squared

00:38 The root of any number (A) squared cancels out the square

00:42 Apply this formula to our exercise and cancel out the squares:

01:02 Break down X squared into factors X and X

01:09 Simplify wherever possible

01:14 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{\frac{100x^4}{25x^2}}=$

Step-by-step solution

Let's solve the problem by following these steps:

Step 1: Simplify the fraction under the square root.
The expression is $\frac{100x^4}{25x^2}$ . We can simplify this by dealing with the coefficient and the variable separately:

The numerical part: $\frac{100}{25} = 4$ .
The variable part: $\frac{x^4}{x^2} = x^{4-2} = x^2$ , using the laws of exponents.

Thus, the simplified fraction is $4x^2$ .

Step 2: Apply the square root.
We have $\sqrt{4x^2}$ . We apply the square root to both terms:

$\sqrt{4} = 2$ , since 2 squared equals 4.
$\sqrt{x^2} = x$ , assuming $x$ is positive (or using the absolute value to ensure non-negativity, but the context suggests a straightforward approach).

Thus, $\sqrt{4x^2} = 2x$ .

Conclusion:

Therefore, the solution to the expression is $2x$ .

Final Answer

$2x$

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify the fraction first, then apply square root
Technique: $\frac{100x^4}{25x^2} = \frac{100}{25} \cdot \frac{x^4}{x^2} = 4x^2$
Check: Verify $(2x)^2 = 4x^2$ matches simplified fraction ✓

Common Mistakes

Avoid these frequent errors

Taking square root of numerator and denominator separately
Don't take $\frac{\sqrt{100x^4}}{\sqrt{25x^2}} = \frac{10x^2}{5x} = 2x$ ! This gives the right answer by luck but misses the proper method. Always simplify the fraction completely first, then apply the square root to the entire simplified expression.

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 the fraction before taking the square root?

Simplifying first makes the problem much easier! $\sqrt{4x^2}$ is simpler to work with than $\sqrt{\frac{100x^4}{25x^2}}$ . Plus, you're less likely to make calculation errors.

How do I divide variables with exponents like x⁴/x²?

Use the quotient rule for exponents: $\frac{x^a}{x^b} = x^{a-b}$ . So $\frac{x^4}{x^2} = x^{4-2} = x^2$ . Just subtract the bottom exponent from the top!

What if x is negative? Does √(x²) still equal x?

Technically, $\sqrt{x^2} = |x|$ (absolute value of x). But in most algebra problems, we assume variables are positive unless stated otherwise, so $\sqrt{x^2} = x$ works fine.

Can I cancel terms directly in the original expression?

Yes! You can cancel $x^2$ from top and bottom: $\frac{100x^4}{25x^2} = \frac{100x^2}{25} = 4x^2$ . This gives the same result as using exponent rules.

How do I check if 2x is the right answer?

Square your answer: $(2x)^2 = 4x^2$ . This should match what you got after simplifying the fraction under the square root. If they're equal, you're correct!

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