Simplify 11^(2a) ÷ 11^5: Exponential Division Problem

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{11^{2a}}{11^5} $ means you have 2a factors of 11 on top and 5 factors of 11 on bottom. When you cancel out 5 factors, you're left with $ 2a - 5 $ factors!

Q: What if the bottom exponent is bigger than the top?

You still subtract! For example, $ \frac{11^3}{11^7} = 11^{3-7} = 11^{-4} $. A negative exponent just means the result is a fraction.

Q: Can I use this rule when the bases are different?

No! This rule only works when the bases are exactly the same. You can't simplify $ \frac{11^3}{7^3} $ using this method.

Q: What if there's a coefficient in front?

Handle coefficients separately! For $ \frac{3 \cdot 11^{2a}}{11^5} $, you get $ 3 \cdot 11^{2a-5} $. The coefficient 3 stays put.

Q: How can I remember when to add vs subtract exponents?

Multiplication = Add exponents: $ 11^2 \cdot 11^3 = 11^{2+3} $Division = Subtract exponents: $ \frac{11^5}{11^2} = 11^{5-2} $

Exponent Division with Variable Powers

Insert the corresponding expression:

$\frac{11^{2a}}{11^5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We will use the formula for dividing powers

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

00:06 equals the number (A) to the power of the difference of exponents (M-N)

00:08 We will use this formula in our exercise

00:10 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:

$\frac{11^{2a}}{11^5}=$

Step-by-step solution

To solve this problem, we apply the Power of a Quotient Rule for Exponents, which states that for any non-zero base $a$ and integers $m$ and $n$ , the expression $\frac{a^m}{a^n} = a^{m-n}$ . In this case, our base $a$ is 11.

Given the expression $\frac{11^{2a}}{11^5}$ , let's simplify it using the rule:

The numerator is $11^{2a}$ .
The denominator is $11^5$ .

Applying the rule:

$\frac{11^{2a}}{11^5} = 11^{2a-5}$

Thus, the expression simplifies to $11^{2a-5}$ .

So, the solution to the question is: $11^{2a-5}$

Final Answer

$11^{2a-5}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Subtract bottom exponent from top: $2a - 5$
Check: If $a = 3$ , then $11^1 = \frac{11^6}{11^5}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when dividing
Don't add the exponents like $2a + 5$ = wrong operation! Division means subtraction, not addition of exponents. Always subtract the denominator exponent from the numerator exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{11^{2a}}{11^5}$ means you have 2a factors of 11 on top and 5 factors of 11 on bottom. When you cancel out 5 factors, you're left with $2a - 5$ factors!

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{11^3}{11^7} = 11^{3-7} = 11^{-4}$ . A negative exponent just means the result is a fraction.

Can I use this rule when the bases are different?

No! This rule only works when the bases are exactly the same. You can't simplify $\frac{11^3}{7^3}$ using this method.

What if there's a coefficient in front?

Handle coefficients separately! For $\frac{3 \cdot 11^{2a}}{11^5}$ , you get $3 \cdot 11^{2a-5}$ . The coefficient 3 stays put.

How can I remember when to add vs subtract exponents?

Multiplication = Add exponents: $11^2 \cdot 11^3 = 11^{2+3}$
Division = Subtract exponents: $\frac{11^5}{11^2} = 11^{5-2}$

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