Simplify the Expression: 12⁵ ÷ 12⁸ Using Laws of Exponents

Q: Why do we subtract exponents when dividing?

Think of it as canceling out! When you have $ \frac{12^5}{12^8} $, you're canceling 5 of the 12's from both top and bottom, leaving 3 extra 12's in the denominator.

Q: What does a negative exponent mean?

A negative exponent means "flip it"! $ 12^{-3} = \frac{1}{12^3} $. The negative sign tells you the power goes in the denominator instead of the numerator.

Q: Are 12^-3 and 1/12^3 really the same thing?

Yes, they're identical! $ 12^{-3} $ is just the compact way to write $ \frac{1}{12^3} $. Both expressions have the exact same value.

Q: When do I get negative exponents?

You get negative exponents when the bottom exponent is larger than the top exponent. Like $ \frac{12^5}{12^8} $ gives 12^(5-8) = 12^(-3).

Q: Do I need to simplify 12^3 in the denominator?

Usually no! Unless specifically asked, leave it as $ \frac{1}{12^3} $. Computing 12³ = 1,728 makes numbers unnecessarily large and harder to work with.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's start.

00:09 We'll use a formula for dividing powers.

00:13 Any base, A, to the power of N, divided by the same base, A, to the power of M,

00:20 equals A to the power of M minus N.

00:24 We'll apply this in our exercise.

00:26 Now, let's use the formula for negative powers.

00:30 Any base, A, to the negative power of N,

00:35 equals one over A to the positive power of N.

00:39 We'll use this in our exercise too.

00:43 We'll substitute one over A and the opposite power.

00:47 And that's how we solve the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\frac{12^5}{12^8}=$

2

Step-by-step solution

To simplify the expression $\frac{12^5}{12^8}$ , we'll follow these steps:

Step 1: Apply the quotient rule for exponents.
Step 2: Simplify and interpret the result using negative exponents if necessary.

Let's work through each step:

Step 1: Apply the quotient rule for exponents.
We are given the expression $\frac{12^5}{12^8}$ . According to the quotient rule for exponents, $\frac{a^m}{a^n} = a^{m-n}$ , so we have:

\frac{12^5}{12^8} = 12^{5-8} = 12^{-3}

Step 2: Simplify and interpret.
The result $12^{-3}$ can be expressed using the concept of negative exponents $a^{-n} = \frac{1}{a^n}$ :

12^{-3} = \frac{1}{12^3}

Therefore, both expressions $12^{-3}$ and $\frac{1}{12^3}$ are equivalent.

Matching with the provided choices:
- Choice 1: $12^{-3}$ - This matches our first result.
- Choice 2: $\frac{1}{12^3}$ - This matches our interpretation of the negative exponent.

Choice 4 states: "a'+b' are correct," which refers to both expressions being correct representations. Therefore, the correct answer is "a'+b' are correct."

3

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{12^5}{12^8} = 12^{5-8} = 12^{-3}$
Check: Convert negative exponent: $12^{-3} = \frac{1}{12^3}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents when dividing: 12^5 ÷ 12^8 ≠ 12^13! This confuses division with multiplication. Always subtract the bottom exponent from the top exponent when dividing same bases.

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 as canceling out! When you have $\frac{12^5}{12^8}$ , you're canceling 5 of the 12's from both top and bottom, leaving 3 extra 12's in the denominator.

What does a negative exponent mean?

+

A negative exponent means "flip it"! $12^{-3} = \frac{1}{12^3}$ . The negative sign tells you the power goes in the denominator instead of the numerator.

Are 12^-3 and 1/12^3 really the same thing?

+

Yes, they're identical! $12^{-3}$ is just the compact way to write $\frac{1}{12^3}$ . Both expressions have the exact same value.

When do I get negative exponents?

+

You get negative exponents when the bottom exponent is larger than the top exponent. Like $\frac{12^5}{12^8}$ gives 12^(5-8) = 12^(-3).

Do I need to simplify 12^3 in the denominator?

+

Usually no! Unless specifically asked, leave it as $\frac{1}{12^3}$ . Computing 12³ = 1,728 makes numbers unnecessarily large and harder to work with.

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Simplify the Expression: 12⁵ ÷ 12⁸ Using Laws of Exponents

Quotient Rule with Negative Exponents

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