Simplify the Square Root: √(x^20/x^24) - Step-by-Step Solution

Q: Why do I get a negative exponent when I subtract 20 - 24?

That's expected! When the denominator has a larger exponent than the numerator, you get a negative exponent. $ x^{20-24} = x^{-4} $ is correct and will simplify nicely.

Q: How does √(x^(-4)) become x^(-2)?

The square root is the same as raising to the 1/2 power. So $ \sqrt{x^{-4}} = (x^{-4})^{1/2} $. Using the power rule: multiply exponents to get $ x^{-4 \times 1/2} = x^{-2} $.

Q: Why is x^(-2) the same as 1/x^2?

Negative exponents mean "reciprocal": $ x^{-n} = \frac{1}{x^n} $. So $ x^{-2} = \frac{1}{x^2} $. This rule helps convert between forms!

Q: Can I check my answer by plugging in a number?

Absolutely! Try x = 3: $ \sqrt{\frac{3^{20}}{3^{24}}} = \sqrt{3^{-4}} = \sqrt{\frac{1}{81}} = \frac{1}{9} $. And $ \frac{1}{x^2} = \frac{1}{3^2} = \frac{1}{9} $ ✓

Radical Simplification with Negative Exponents

Solve the following exercise:

$\sqrt{\frac{x^{20}}{x^{24}}}=$

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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 It can be written as root of the numerator (A) divided by root of the denominator (B)

00:09 Apply this formula to our exercise

00:20 When we have a root of the order (B) on number (X) to the power of (A)

00:24 The result equals the number (X) to the power of (A divided by B)

00:33 The "regular" root is of the order 2

00:37 We will apply this formula to our exercise

00:48 Calculate the power quotients

00:56 Break down X to the power of 12 into factors of X to the power of 10 and X squared

01:05 Reduce wherever possible

01:08 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:

$\sqrt{\frac{x^{20}}{x^{24}}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Simplify the expression inside the square root using exponent rules.
Evaluate the square root of the simplified expression.

Let's work through the solution step-by-step:

First, simplify the expression inside the square root. We have:

$\frac{x^{20}}{x^{24}} = x^{20-24} = x^{-4}$

Next, apply the square root to the simplified expression:

$\sqrt{x^{-4}} = \left(x^{-4}\right)^{1/2} = x^{-4 \times \frac{1}{2}} = x^{-2}$

This can be written as:

$x^{-2} = \frac{1}{x^2}$

Therefore, the solution to the problem is $\frac{1}{x^2}$ .

Final Answer

$\frac{1}{x^2}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: Subtract exponents when dividing same bases: x^m/x^n = x^(m-n)
Square Root: Convert to fractional exponent: √(x^(-4)) = x^(-4×1/2) = x^(-2)
Check: Verify x^(-2) = 1/x^2 by substituting x = 2: 1/4 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to use quotient rule first
Don't try to take the square root of each term separately like √(x^20)/√(x^24) = x^10/x^12 = wrong answer! This ignores proper order of operations and gives x^(-2) instead of the correct 1/x^2. Always simplify inside the radical first using quotient rule: x^20/x^24 = x^(-4), then apply the square root.

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 get a negative exponent when I subtract 20 - 24?

That's expected! When the denominator has a larger exponent than the numerator, you get a negative exponent. $x^{20-24} = x^{-4}$ is correct and will simplify nicely.

How does √(x^(-4)) become x^(-2)?

The square root is the same as raising to the 1/2 power. So $\sqrt{x^{-4}} = (x^{-4})^{1/2}$ . Using the power rule: multiply exponents to get $x^{-4 \times 1/2} = x^{-2}$ .

Why is x^(-2) the same as 1/x^2?

Negative exponents mean "reciprocal": $x^{-n} = \frac{1}{x^n}$ . So $x^{-2} = \frac{1}{x^2}$ . This rule helps convert between forms!

Can I check my answer by plugging in a number?

Absolutely! Try x = 3: $\sqrt{\frac{3^{20}}{3^{24}}} = \sqrt{3^{-4}} = \sqrt{\frac{1}{81}} = \frac{1}{9}$ . And $\frac{1}{x^2} = \frac{1}{3^2} = \frac{1}{9}$ ✓

What if x is negative? Does this still work?

Be careful with negative bases and even roots! This problem assumes x > 0 to avoid complex numbers. When x is positive, all our steps work perfectly.

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