Square Root Simplification with Fractional Exponents

Q: Why do I subtract the exponents in the fraction?

The quotient rule states that $ \frac{x^m}{x^n} = x^{m-n} $. When you divide powers with the same base, you subtract the bottom exponent from the top exponent.

Q: Can I take the square root of each part separately?

While mathematically possible, it's much easier to simplify first! $ \sqrt{\frac{x^8}{x^4}} $ becomes $ \sqrt{x^4} $, which is simpler than working with $ \frac{\sqrt{x^8}}{\sqrt{x^4}} $.

Q: How does the square root become an exponent of 1/2?

Square roots and fractional exponents are equivalent: $ \sqrt{x} = x^{1/2} $. So $ \sqrt{x^4} = (x^4)^{1/2} = x^{4 \cdot 1/2} = x^2 $ using the power rule.

Q: What if the variable could be negative?

For this type of problem, we typically assume positive values for the variable. With negative values, we'd need to consider absolute value: $ \sqrt{x^4} = |x^2| = x^2 $ since $ x^2 $ is always positive.

Q: Do I always subtract exponents when I see division?

Only when the bases are the same! $ \frac{x^5}{x^3} = x^2 $ works, but $ \frac{x^5}{y^3} $ cannot be simplified this way because the bases are different.

Exponent Rules with Square Root Operations

Solve the following exercise:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this problem together.

00:13 When you have a root of a fraction, like A over B.

00:17 You can think of it as the root of A, over the root of B.

00:22 Now, let's use this idea in our exercise.

00:27 First, change X to the power of 8, into X to the power of 4 squared.

00:32 Next, turn X to the power of 4, into X squared squared.

00:38 Remember, the root of A squared, cancels out the square.

00:43 Now, apply this to cancel the squares in our exercise.

00:55 Break X to the power of 4 into X squared times X squared.

01:00 Simplify everything you can.

01:03 And that's our solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

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

Step-by-step solution

To solve this problem, we'll simplify the given expression step by step.

Firstly, observe the expression: $\sqrt{\frac{x^8}{x^4}}$ .

Step 1: Apply the quotient of powers rule: The expression inside the square root is $\frac{x^8}{x^4}$ , which simplifies to $x^{8-4} = x^4$ using the rule $\frac{x^m}{x^n} = x^{m-n}$ .
Step 2: Apply the square root rule: Now we have $\sqrt{x^4}$ . Utilizing the property of square roots, we find $\sqrt{x^4} = x^{4/2} = x^2$ .

Therefore, the simplified expression is $\textbf{x}^2$ .

Thus, the final solution to the problem is $\textbf{x}^2$ , which corresponds to choice 2 in the given list of options.

Final Answer

$x^2$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{x^8}{x^4} = x^{8-4} = x^4$ before taking square root
Check: Verify $(x^2)^2 = x^4$ matches our simplified expression ✓

Common Mistakes

Avoid these frequent errors

Taking square root before simplifying the fraction
Don't try to find $\sqrt{x^8}$ and $\sqrt{x^4}$ separately = $\frac{x^4}{x^2} = x^2$ ! This makes the problem unnecessarily complex and prone to errors. Always simplify inside the radical first using exponent rules.

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 subtract the exponents in the fraction?

The quotient rule states that $\frac{x^m}{x^n} = x^{m-n}$ . When you divide powers with the same base, you subtract the bottom exponent from the top exponent.

Can I take the square root of each part separately?

While mathematically possible, it's much easier to simplify first! $\sqrt{\frac{x^8}{x^4}}$ becomes $\sqrt{x^4}$ , which is simpler than working with $\frac{\sqrt{x^8}}{\sqrt{x^4}}$ .

How does the square root become an exponent of 1/2?

Square roots and fractional exponents are equivalent: $\sqrt{x} = x^{1/2}$ . So $\sqrt{x^4} = (x^4)^{1/2} = x^{4 \cdot 1/2} = x^2$ using the power rule.

What if the variable could be negative?

For this type of problem, we typically assume positive values for the variable. With negative values, we'd need to consider absolute value: $\sqrt{x^4} = |x^2| = x^2$ since $x^2$ is always positive.

Do I always subtract exponents when I see division?

Only when the bases are the same! $\frac{x^5}{x^3} = x^2$ works, but $\frac{x^5}{y^3}$ cannot be simplified this way because the bases are different.

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