Evaluate the Exponential Fraction: 2^9 ÷ 11^9

Q: Why can't I just multiply the exponents by the fraction?

Exponents don't work that way! The expression $ \frac{2^9}{11^9} $ means 2 multiplied by itself 9 times, divided by 11 multiplied by itself 9 times. We're not multiplying 9 by the fraction.

Q: How do I remember this exponent rule?

Think of it as factoring out the common exponent. Just like $ 3x + 5x = (3+5)x $, we have $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $ where the n is factored out!

Q: What if the exponents were different numbers?

If the exponents were different, like $ \frac{2^9}{11^7} $, you cannot use this rule! The exponents must be identical for $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $ to work.

Q: Can I calculate the actual numerical value?

Yes, but it's not necessary here! $ 2^9 = 512 $ and $ 11^9 $ is huge, so $ \left(\frac{2}{11}\right)^9 $ is the simplest form and exactly what the question asks for.

Q: Why is this form better than leaving it as a division?

The form $ \left(\frac{2}{11}\right)^9 $ is more compact and clear. It shows the relationship between the base fraction and the power, making it easier to work with in further calculations.

Exponential Fractions with Same Power

Insert the corresponding expression:

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

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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, each raised to the same power (N)

00:11 We will apply this formula to our exercise, only this time in the opposite direction

00:19 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:

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

Step-by-step solution

To solve this problem, we'll employ the exponent rules for fractions:

Step 1: Recognize that $\frac{2^9}{11^9}$ follows the general form $\frac{a^n}{b^n}$ .
Step 2: Apply the property $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
Step 3: Substitute into the property to express the fraction as $\left(\frac{2}{11}\right)^9$ .

Let's work through the steps in detail:

Step 1: The expression $\frac{2^9}{11^9}$ can be viewed as each number, 2 and 11, raised to the 9th power in a fraction.

Step 2: Utilize the exponent rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to rewrite the fraction with a single power.

Step 3: Therefore, the expression $\frac{2^9}{11^9}$ simplifies to $\left(\frac{2}{11}\right)^9$ .

Therefore, the correct answer is indeed $\left(\frac{2}{11}\right)^9$ .

The correct choice from the provided options is:

$\left(\frac{2}{11}\right)^9$

Final Answer

$\left(\frac{2}{11}\right)^9$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same exponent, use $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$
Technique: Transform $\frac{2^9}{11^9}$ into $\left(\frac{2}{11}\right)^9$ using the property
Check: Verify that $\left(\frac{2}{11}\right)^9 = \frac{2^9}{11^9}$ by expanding back ✓

Common Mistakes

Avoid these frequent errors

Applying exponent only to numerator or denominator
Don't write $\frac{2^9}{11^9}$ as $\left(\frac{2}{11}\right)^{\frac{1}{9}}$ or multiply exponents incorrectly = wrong base or power! This misapplies the exponent rules and creates completely different values. Always recognize that identical exponents on numerator and denominator combine as $\left(\frac{a}{b}\right)^n$ .

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 multiply the exponents by the fraction?

Exponents don't work that way! The expression $\frac{2^9}{11^9}$ means 2 multiplied by itself 9 times, divided by 11 multiplied by itself 9 times. We're not multiplying 9 by the fraction.

How do I remember this exponent rule?

Think of it as factoring out the common exponent. Just like $3x + 5x = (3+5)x$ , we have $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ where the n is factored out!

What if the exponents were different numbers?

If the exponents were different, like $\frac{2^9}{11^7}$ , you cannot use this rule! The exponents must be identical for $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ to work.

Can I calculate the actual numerical value?

Yes, but it's not necessary here! $2^9 = 512$ and $11^9$ is huge, so $\left(\frac{2}{11}\right)^9$ is the simplest form and exactly what the question asks for.

Why is this form better than leaving it as a division?

The form $\left(\frac{2}{11}\right)^9$ is more compact and clear. It shows the relationship between the base fraction and the power, making it easier to work with in further calculations.

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