Evaluate (6/11×13×15)^xy: Complex Fraction with Variable Exponents

Q: Why are both forms B and C correct?

Both expressions are mathematically equivalent! Form B: $ \frac{6^{xy}}{11^{xy}×13^{xy}×15^{xy}} $ shows the distributed version, while Form C: $ \frac{6^{xy}}{(11×13×15)^{xy}} $ keeps the denominator grouped. They represent the same value.

Q: When should I distribute the exponent?

It depends on what you need to do next! Distributed form is helpful for simplifying individual factors, while grouped form is better for seeing the overall structure. Both are correct.

Q: What if the exponent was just a number instead of xy?

The same rules apply! For example: $ \left(\frac{6}{11×13×15}\right)^3 = \frac{6^3}{(11×13×15)^3} $ or $ \frac{6^3}{11^3×13^3×15^3} $

Q: Can I simplify this expression further?

You could calculate $ 11×13×15 = 2145 $ to get $ \frac{6^{xy}}{2145^{xy}} $, but usually the factored form is more useful for further algebraic work.

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{6}{11\times13\times15}\right)^{xy}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's simplify this problem together!

00:15 Remember, according to the exponent rules, when a fraction is raised to the power of N,

00:21 both the top and bottom of the fraction are raised to this power.

00:25 We'll use this idea in our exercise.

00:38 Now, when a product is raised to the power of N,

00:42 each part of the product is raised to this same power.

00:48 We're going to apply this step to our example.

00:58 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{6}{11\times13\times15}\right)^{xy}=$

Step-by-step solution

First, let's apply the exponent to the entire fraction:

$\left(\frac{6}{11 \times 13 \times 15}\right)^{xy} = \frac{6^{xy}}{(11 \times 13 \times 15)^{xy}}$

Now, distribute the $xy$ exponent in the denominator to each factor:

$\frac{6^{xy}}{11^{xy} \times 13^{xy} \times 15^{xy}}$

Thus, the rewritten expression is $\frac{6^{xy}}{11^{xy} \times 13^{xy} \times 15^{xy}}$ .

Comparing our expression with the options given and based on our simplification, option 3: $\frac{6^{xy}}{\left(11 \times 13 \times 15\right)^{xy}}$ makes sense, as well as option 2: $\frac{6^{xy}}{11^{xy}\times13^{xy}\times15^{xy}}$ after distributing the exponent within the denominator.

Therefore, both options B and C are correct, making the right choice option 4.

Therefore, the correct answer is: B+C are correct.

Final Answer

B+C are correct

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Distribution: $(abc)^n = a^n \times b^n \times c^n$ for products
Check: Both forms $\frac{6^{xy}}{(11×13×15)^{xy}}$ and distributed version are equivalent ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply exponent to numerator
Don't leave 6 without the xy exponent = $\frac{6}{(11×13×15)^{xy}}$ ! This ignores the fundamental rule that exponents apply to the entire fraction. Always raise both numerator and denominator to the given power: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are both forms B and C correct?

Both expressions are mathematically equivalent! Form B: $\frac{6^{xy}}{11^{xy}×13^{xy}×15^{xy}}$ shows the distributed version, while Form C: $\frac{6^{xy}}{(11×13×15)^{xy}}$ keeps the denominator grouped. They represent the same value.

When should I distribute the exponent?

It depends on what you need to do next! Distributed form is helpful for simplifying individual factors, while grouped form is better for seeing the overall structure. Both are correct.

What if the exponent was just a number instead of xy?

The same rules apply! For example: $\left(\frac{6}{11×13×15}\right)^3 = \frac{6^3}{(11×13×15)^3}$ or $\frac{6^3}{11^3×13^3×15^3}$

How do I know which form to choose?

Look at what the question asks for! If multiple forms are given as options and they're mathematically equivalent, both can be correct. Always check if the question says 'select all that apply' or similar.

Can I simplify this expression further?

You could calculate $11×13×15 = 2145$ to get $\frac{6^{xy}}{2145^{xy}}$ , but usually the factored form is more useful for further algebraic work.

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