Simplify (6/xy)²: Converting Squared Fraction Expression

Q: Why can't I just write 6² in the numerator and leave the denominator as xy?

When you raise a fraction to a power, the exponent affects both the numerator and denominator. Think of it like this: $ \left(\frac{a}{b}\right)^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2} $

Q: What's the difference between (xy)² and x²y²?

They're actually the same thing! $ (xy)^2 = x^2y^2 $ because when you multiply variables with exponents, you add the exponents: $ x \cdot x \cdot y \cdot y = x^2y^2 $

Q: How do I know which answer choice shows the correct format?

Look for the answer that shows both parts squared: the numerator should be $ 6^2 = 36 $ and the denominator should be $ (xy)^2 $ or $ x^2y^2 $

Q: Can I simplify (xy)² further?

Yes! You can write $ (xy)^2 $ as $ x^2y^2 $, but both forms are correct. The parentheses version $ (xy)^2 $ clearly shows what was squared.

Q: What if I had different numbers or variables?

The same rule applies! For example: $ \left(\frac{4}{ab}\right)^3 = \frac{4^3}{(ab)^3} = \frac{64}{a^3b^3} $. Always raise both numerator and denominator to the given power.

Fraction Exponents with Variable Denominators

Insert the corresponding expression:

$\left(\frac{6}{x\times y}\right)^2=$

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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 According to the laws of exponents, a fraction raised to the power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:11 We will apply this formula to our exercise

00:16 We'll raise both the numerator and the denominator to the power (N)

00:19 We'll calculate 6 squared and then substitute it into the expression

00:24 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{6}{x\times y}\right)^2=$

Step-by-step solution

To solve this problem, we'll apply the rule for powers of a fraction:

Step 1: Identify the fraction. The fraction given is $\frac{6}{x \times y}$ .
Step 2: Apply the power to both the numerator and the denominator. This means squaring both 6 and $x \times y$ .
Step 3: Calculate the square of the numerator and the denominator:

The square of the numerator: $6^2 = 36$ .
The square of the denominator: $(x \times y)^2 = x^2 \times y^2$ .

Step 4: Combine the results: $\left(\frac{6}{x \times y}\right)^2 = \frac{6^2}{(x \times y)^2} = \frac{36}{x^2 \times y^2}$ .

Thus, the expression $\left(\frac{6}{x \times y}\right)^2$ simplifies to $\frac{36}{(x \times y)^2}$ .

Therefore, the correct answer is clearly the expression $\frac{36}{(x \times y)^2}$ , which matches choice 4.

Final Answer

$\frac{36}{\left(x\times y\right)^2}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: Square 6 to get 36, square (xy) to get (xy)²
Check: Verify that $\frac{36}{(xy)^2}$ equals the original expression squared ✓

Common Mistakes

Avoid these frequent errors

Only squaring the numerator
Don't square just the 6 and leave xy unchanged = $\frac{36}{xy}$ ! This ignores the exponent rule for fractions and gives an incorrect result. Always apply the exponent to both the top AND bottom of the fraction.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just write 6² in the numerator and leave the denominator as xy?

When you raise a fraction to a power, the exponent affects both the numerator and denominator. Think of it like this: $\left(\frac{a}{b}\right)^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2}$

What's the difference between (xy)² and x²y²?

They're actually the same thing! $(xy)^2 = x^2y^2$ because when you multiply variables with exponents, you add the exponents: $x \cdot x \cdot y \cdot y = x^2y^2$

How do I know which answer choice shows the correct format?

Look for the answer that shows both parts squared: the numerator should be $6^2 = 36$ and the denominator should be $(xy)^2$ or $x^2y^2$

Can I simplify (xy)² further?

Yes! You can write $(xy)^2$ as $x^2y^2$ , but both forms are correct. The parentheses version $(xy)^2$ clearly shows what was squared.

What if I had different numbers or variables?

The same rule applies! For example: $\left(\frac{4}{ab}\right)^3 = \frac{4^3}{(ab)^3} = \frac{64}{a^3b^3}$ . Always raise both numerator and denominator to the given power.

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