Simplify the Complex Fraction: (2⁹×3⁹)/(11⁹×7⁹)

Q: Why can I combine 2^9 and 3^9 but not 2^9 and 7^9?

You can combine any powers with the same exponent! The rule $ a^n \times b^n = (ab)^n $ works for all numbers. So $ 2^9 \times 3^9 = (2 \times 3)^9 $ and $ 11^9 \times 7^9 = (11 \times 7)^9 $.

Q: Which answer choice is actually correct?

Both choices a' and b' are mathematically equivalent! $ \frac{(2 \times 3)^9}{(11 \times 7)^9} $ and $ \left(\frac{2 \times 3}{11 \times 7}\right)^9 $ represent the same value, just written differently using the quotient rule for exponents.

Q: How do I know when to use the quotient rule vs keeping fractions separate?

Use the quotient rule $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $ when you want a simpler form. Both forms are correct, but $ \left(\frac{6}{77}\right)^9 $ is more compact than $ \frac{6^9}{77^9} $.

Q: What's the final numerical value?

The expression simplifies to $ \left(\frac{6}{77}\right)^9 $, which equals approximately 0.000000456. But for this problem, the algebraic form is what matters most!

Q: Can I cancel terms in the original expression?

No! You cannot cancel $ 2^9 $ with $ 11^9 $ because they have different bases. Only identical factors can be canceled. First combine like exponents, then simplify the resulting fraction.

Exponent Rules with Fraction Simplification

Insert the corresponding expression:

$\frac{2^9\times3^9}{11^9\times7^9}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:03 According to the laws of exponents, a product raised to the power (N)

00:08 equals a product broken down into factors where each factor is raised to the power (N)

00:12 We'll apply this formula to our exercise, converting to parentheses with a power

00:19 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)

00:28 Now we'll apply the second formula and convert the expression to a fraction inside of parentheses raised to a power

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:

$\frac{2^9\times3^9}{11^9\times7^9}=$

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases are multiplied with same exponent, combine first
Technique: $2^9 \times 3^9 = (2 \times 3)^9 = 6^9$
Check: Both forms equal $\left(\frac{6}{77}\right)^9$ when simplified ✓

Common Mistakes

Avoid these frequent errors

Combining exponents instead of bases
Don't add exponents like 2^9 × 3^9 = 5^18! This ignores that multiplication with same exponents means combining bases first. Always use (ab)^n = a^n × b^n in reverse: a^n × b^n = (ab)^n.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine 2^9 and 3^9 but not 2^9 and 7^9?

You can combine any powers with the same exponent! The rule $a^n \times b^n = (ab)^n$ works for all numbers. So $2^9 \times 3^9 = (2 \times 3)^9$ and $11^9 \times 7^9 = (11 \times 7)^9$ .

Which answer choice is actually correct?

Both choices a' and b' are mathematically equivalent! $\frac{(2 \times 3)^9}{(11 \times 7)^9}$ and $\left(\frac{2 \times 3}{11 \times 7}\right)^9$ represent the same value, just written differently using the quotient rule for exponents.

How do I know when to use the quotient rule vs keeping fractions separate?

Use the quotient rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ when you want a simpler form. Both forms are correct, but $\left(\frac{6}{77}\right)^9$ is more compact than $\frac{6^9}{77^9}$ .

What's the final numerical value?

The expression simplifies to $\left(\frac{6}{77}\right)^9$ , which equals approximately 0.000000456. But for this problem, the algebraic form is what matters most!

Can I cancel terms in the original expression?

No! You cannot cancel $2^9$ with $11^9$ because they have different bases. Only identical factors can be canceled. First combine like exponents, then simplify the resulting fraction.

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