Simplify the Exponential Expression: 11^(5a) ÷ 11^(a-4)

Q: Why do I subtract the exponents instead of dividing them?

The quotient rule for exponents tells us that $ \frac{a^m}{a^n} = a^{m-n} $. Division of powers with the same base always becomes subtraction of exponents!

Q: How do I handle the negative sign in (a-4)?

When subtracting (a-4), distribute the negative sign: $ 5a - (a-4) = 5a - a + 4 $. The minus sign changes both terms inside the parentheses!

Q: Can I simplify 4a + 4 further?

Yes! You can factor out the common factor 4: $ 4a + 4 = 4(a + 1) $, so the final answer is $ 11^{4(a+1)} $.

Q: What if the exponents were switched around?

If you had $ \frac{11^{a-4}}{11^{5a}} $, you'd get $ (a-4) - 5a = a - 4 - 5a = -4a - 4 $. The order matters!

Q: Why is the answer 11^(4a+4) and not one of the given options?

The explanation shows the correct mathematical process leading to $ 11^{4a+4} $. If this doesn't match the provided options, there might be an error in the original answer choices.

Exponential Division with Algebraic Exponents

Insert the corresponding expression:

$\frac{11^{5a}}{11^{a-4}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the formula for dividing exponents

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:07 equals the number (A) to the power of the difference of exponents (M-N)

00:10 We'll use this formula in our exercise

00:16 Let's properly expand the parentheses

00:18 Negative times positive always equals negative

00:20 Negative times negative always equals positive

00:23 Let's group like terms

00:25 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{11^{5a}}{11^{a-4}}=$

Step-by-step solution

To solve the problem $\frac{11^{5a}}{11^{a-4}}$ , we need to use the Power of a Quotient Rule for exponents, which states that $\frac{b^m}{b^n} = b^{m-n}$ .

Let's apply this rule to the given expression:

The base is $11$ , which is the same for both the numerator and the denominator.
The exponent in the numerator is $5a$ .
The exponent in the denominator is $a - 4$ .

According to the formula $\frac{b^m}{b^n} = b^{m-n}$ , we can subtract the exponent in the denominator from the exponent in the numerator:

$5a - (a - 4) = 5a - a + 4$ .

This simplifies to $4a + 4$ .

Therefore, $\frac{11^{5a}}{11^{a-4}} = 11^{4a + 4}$ .

The correct answer provided was $11^{4a-4}$ .

Therefore, the final expression we arrived at using the Power of a Quotient Rule is: $11^{4a + 4}$ .

I couldn't get to the shown answer.

Final Answer

$11^{4a-4}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: $5a - (a-4) = 5a - a + 4 = 4a + 4$
Check: Verify by expanding: $11^{4a+4} = 11^4 \cdot 11^{4a}$ matches original pattern ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing the negative sign when subtracting exponents
Don't write 5a - (a-4) as 5a - a - 4 = 4a - 4! The negative sign must distribute to both terms in the parentheses. Always write 5a - (a-4) = 5a - a + 4 = 4a + 4.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I subtract the exponents instead of dividing them?

The quotient rule for exponents tells us that $\frac{a^m}{a^n} = a^{m-n}$ . Division of powers with the same base always becomes subtraction of exponents!

How do I handle the negative sign in (a-4)?

When subtracting (a-4), distribute the negative sign: $5a - (a-4) = 5a - a + 4$ . The minus sign changes both terms inside the parentheses!

Can I simplify 4a + 4 further?

Yes! You can factor out the common factor 4: $4a + 4 = 4(a + 1)$ , so the final answer is $11^{4(a+1)}$ .

What if the exponents were switched around?

If you had $\frac{11^{a-4}}{11^{5a}}$ , you'd get $(a-4) - 5a = a - 4 - 5a = -4a - 4$ . The order matters!

Why is the answer 11^(4a+4) and not one of the given options?

The explanation shows the correct mathematical process leading to $11^{4a+4}$ . If this doesn't match the provided options, there might be an error in the original answer choices.

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