Simplify the Complex Fraction: (6⁶ × 11⁶)/(5⁶ × 13⁶)

Q: Why can I combine 6⁶ × 11⁶ but not 6⁶ × 5⁶?

You can only use $ a^m \times b^m = (a \times b)^m $ when the exponents are identical. Here, 6⁶ × 11⁶ both have exponent 6, so they combine. But 6⁶ × 5⁶ are in different parts of the fraction!

Q: How do I know which terms to group together?

Keep numerator and denominator separate! Group all numerator terms: $ 6^6 \times 11^6 $, then group all denominator terms: $ 5^6 \times 13^6 $. Never mix across the fraction line.

Q: Can I simplify 66⁶/(65⁶) any further?

The final form $ \left(\frac{66}{65}\right)^6 $ is the most simplified using power laws. You could calculate the decimal value, but the exponential form shows the mathematical structure clearly.

Q: What if the exponents were different numbers?

If exponents don't match, you cannot use the combining rule! For example, $ 6^5 \times 11^6 $ stays as separate terms. The power law $ a^m \times b^m = (ab)^m $ only works when exponents are identical.

Q: Why are both options B and C correct?

Option B shows $ \frac{(6 \times 11)^6}{(5 \times 13)^6} $ and option C shows $ \left(\frac{6 \times 11}{5 \times 13}\right)^6 $. These are mathematically equivalent by the quotient power rule: $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $!

Power Laws with Complex Fractions

Insert the corresponding expression:

$\frac{6^6\times11^6}{5^6\times13^6}=$

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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 product raised to the power (N)

00:07 equals the product broken down into factors where each factor is raised to the power (N)

00:11 We'll apply this formula to our exercise, converting to parentheses with an exponent

00:21 According to the laws of exponents, a fraction raised to the power (N)

00:24 equals the numerator and denominator, each raised to the same power (N)

00:30 Now we'll apply the second formula and convert the expression into a fraction within parentheses with an exponent

00:34 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:

$\frac{6^6\times11^6}{5^6\times13^6}=$

Step-by-step solution

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

Step 1: Apply the product of powers formula
Step 2: Simplify the combined powers into a single fraction with one exponent

Now, let's work through each step:
Step 1: Using $(a^m \times b^m) = (a \times b)^m$ , we get:
$(6^6 \times 11^6) = (6 \times 11)^6$ and $(5^6 \times 13^6) = (5 \times 13)^6$ .

Step 2: Simplifying the expression, we get:
$\frac{(6^6 \times 11^6)}{(5^6 \times 13^6)} = \frac{(6 \times 11)^6}{(5 \times 13)^6} = \left(\frac{6 \times 11}{5 \times 13}\right)^6$ .

This transformation matches both option B and C in the provided answer choices.

Therefore, the correct answer is $\textbf{B + C are correct}$ .

Final Answer

B+C are correct

Key Points to Remember

Essential concepts to master this topic

Power Law: When bases have same exponent, combine then raise to power
Technique: $6^6 \times 11^6 = (6 \times 11)^6 = 66^6$
Check: Verify both numerator and denominator follow same pattern ✓

Common Mistakes

Avoid these frequent errors

Applying power laws incorrectly to mixed operations
Don't try to combine different bases with different exponents like $6^6 \times 5^6$ = something wrong! This mixes numerator and denominator incorrectly. Always group same positions: numerators together $(6 \times 11)^6$ , denominators together $(5 \times 13)^6$ .

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why can I combine 6⁶ × 11⁶ but not 6⁶ × 5⁶?

You can only use $a^m \times b^m = (a \times b)^m$ when the exponents are identical. Here, 6⁶ × 11⁶ both have exponent 6, so they combine. But 6⁶ × 5⁶ are in different parts of the fraction!

How do I know which terms to group together?

Keep numerator and denominator separate! Group all numerator terms: $6^6 \times 11^6$ , then group all denominator terms: $5^6 \times 13^6$ . Never mix across the fraction line.

Can I simplify 66⁶/(65⁶) any further?

The final form $\left(\frac{66}{65}\right)^6$ is the most simplified using power laws. You could calculate the decimal value, but the exponential form shows the mathematical structure clearly.

What if the exponents were different numbers?

If exponents don't match, you cannot use the combining rule! For example, $6^5 \times 11^6$ stays as separate terms. The power law $a^m \times b^m = (ab)^m$ only works when exponents are identical.

Why are both options B and C correct?

Option B shows $\frac{(6 \times 11)^6}{(5 \times 13)^6}$ and option C shows $\left(\frac{6 \times 11}{5 \times 13}\right)^6$ . These are mathematically equivalent by the quotient power rule: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ !

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