Simplify Square Root Expression: √(x⁸/x¹⁰)

Q: Why do I subtract 8 - 10 instead of 10 - 8?

The order matters! Since we have $ \frac{x^8}{x^{10}} $, we calculate the exponent as 8 - 10 = -2. If it were $ \frac{x^{10}}{x^8} $, then we'd do 10 - 8 = 2.

Q: What does a negative exponent mean?

A negative exponent means "one over" the positive exponent. So $ x^{-2} = \frac{1}{x^2} $ and $ x^{-1} = \frac{1}{x} $. It's not a negative number!

Q: How do I take the square root of something with a negative exponent?

Use the power rule: $ \sqrt{x^{-2}} = (x^{-2})^{1/2} = x^{-2 \cdot 1/2} = x^{-1} $. The square root multiplies the exponent by 1/2.

Q: Can I simplify this a different way?

Yes! You could rewrite it as $ \sqrt{\frac{x^8}{x^{10}}} = \frac{\sqrt{x^8}}{\sqrt{x^{10}}} = \frac{x^4}{x^5} = x^{-1} = \frac{1}{x} $. Same answer!

Q: What if x is negative?

For this problem, we assume x > 0 since we're dealing with square roots. If x could be negative, we'd need absolute value signs to ensure the result is real.

Radical Simplification with Negative Exponents

Solve the following exercise:

$\sqrt{\frac{x^8}{x^{10}}}=$

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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 is a root of a fraction (A divided by B)

00:06 We can write it as the root of the numerator (A) divided by the root of the denominator (B)

00:09 Apply this formula to our exercise

00:14 Break down X to the power of 8 into X to the power of 4 squared

00:20 Break down X to the power of 10 into X to the power of 5 squared

00:25 The root of any number (A) squared cancels out the square

00:29 Apply this formula to our exercise and cancel out the squares

00:40 Break down X to the power of 5 into factors X to the power of 4 and X

00:48 Reduce wherever possible

00:51 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:

$\sqrt{\frac{x^8}{x^{10}}}=$

Step-by-step solution

To solve this mathematical expression, follow these steps:

Step 1: Use the Quotient Property of Exponents

Simplify the expression inside the square root using the rule:
$\frac{x^8}{x^{10}} = x^{8-10} = x^{-2}$ .
Step 2: Apply the Square Root Property

Now, apply the square root:
$\sqrt{x^{-2}} = (x^{-2})^{1/2} = x^{-2 \cdot \frac{1}{2}} = x^{-1}$ .
Step 3: Express in Simpler Form

The expression $x^{-1}$ can be written as $\frac{1}{x}$ .

Therefore, the final simplified form of the expression is $\frac{1}{x}$ .

Final Answer

$\frac{1}{x}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing powers, subtract exponents: x⁸ ÷ x¹⁰ = x⁻²
Technique: Apply square root to negative exponent: $\sqrt{x^{-2}} = x^{-1}$
Check: Verify $\frac{1}{x} \cdot \frac{1}{x} = \frac{1}{x^2} = x^{-2}$ under radical ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting when dividing
Don't add exponents: 8 + 10 = 18 gives $x^{18}$ under the radical = completely wrong! When dividing powers with same base, you subtract exponents, not add them. Always use $\frac{x^a}{x^b} = x^{a-b}$ for division.

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 8 - 10 instead of 10 - 8?

The order matters! Since we have $\frac{x^8}{x^{10}}$ , we calculate the exponent as 8 - 10 = -2. If it were $\frac{x^{10}}{x^8}$ , then we'd do 10 - 8 = 2.

What does a negative exponent mean?

A negative exponent means "one over" the positive exponent. So $x^{-2} = \frac{1}{x^2}$ and $x^{-1} = \frac{1}{x}$ . It's not a negative number!

How do I take the square root of something with a negative exponent?

Use the power rule: $\sqrt{x^{-2}} = (x^{-2})^{1/2} = x^{-2 \cdot 1/2} = x^{-1}$ . The square root multiplies the exponent by 1/2.

Can I simplify this a different way?

Yes! You could rewrite it as $\sqrt{\frac{x^8}{x^{10}}} = \frac{\sqrt{x^8}}{\sqrt{x^{10}}} = \frac{x^4}{x^5} = x^{-1} = \frac{1}{x}$ . Same answer!

What if x is negative?

For this problem, we assume x > 0 since we're dealing with square roots. If x could be negative, we'd need absolute value signs to ensure the result is real.

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