Simplify the Exponential Expression: 5³÷5⁸

Q: Why do I subtract the exponents when dividing?

Division is the opposite of multiplication! When multiplying like bases, you add exponents. So when dividing, you do the opposite and subtract them.

Q: What does a negative exponent mean?

A negative exponent means "one over" the positive version. So $ 5^{-5} = \frac{1}{5^5} $. It's not a negative number!

Q: Can I just cancel out the 5s?

No! You can't cancel the bases - you must use the exponent rule. The bases stay the same (5), but you subtract the exponents (3 - 8 = -5).

Q: How do I remember which exponent to subtract from which?

Think of it as "top minus bottom": $ \frac{5^3}{5^8} = 5^{\text{top} - \text{bottom}} = 5^{3-8} $. The numerator exponent comes first!

Q: Is my answer wrong if I get a negative exponent?

Not at all! Negative exponents are perfectly valid answers. They just represent fractions. $ 5^{-5} $ is the correct simplified form.

Exponential Division with Negative Exponents

Insert the corresponding expression:

$\frac{5^3}{5^8}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's get started.

00:09 We're going to explore dividing powers.

00:12 If you have A raised to the power of N,

00:16 divided by A raised to the power of M,

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

00:24 We'll use this rule in our practice problem.

00:27 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{5^3}{5^8}=$

Step-by-step solution

We need to simplify the expression $\frac{5^3}{5^8}$ using the rules of exponents. Specifically, we will use the power of a quotient rule for exponents which states that when you divide like bases you subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$ .

Here, the base is 5, the exponent in the numerator is 3, and the exponent in the denominator is 8.

Apply the rule: $5^{3-8}$
Subtract the exponents: $5^{-5}$ .

Therefore, the simplified expression is $5^{-5}$ .

The solution to the question is: $5^{-5}$

Final Answer

$5^{-5}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing like bases, subtract the exponents
Technique: Apply $\frac{a^m}{a^n} = a^{m-n}$ to get $5^{3-8} = 5^{-5}$
Check: Verify that $5^{-5} = \frac{1}{5^5} = \frac{1}{3125}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 3 + 8 = 11 to get $5^{11}$ ! Addition is only for multiplication of like bases. Always subtract the bottom exponent from the top exponent when dividing.

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 when dividing?

Division is the opposite of multiplication! When multiplying like bases, you add exponents. So when dividing, you do the opposite and subtract them.

What does a negative exponent mean?

A negative exponent means "one over" the positive version. So $5^{-5} = \frac{1}{5^5}$ . It's not a negative number!

Can I just cancel out the 5s?

No! You can't cancel the bases - you must use the exponent rule. The bases stay the same (5), but you subtract the exponents (3 - 8 = -5).

How do I remember which exponent to subtract from which?

Think of it as "top minus bottom": $\frac{5^3}{5^8} = 5^{\text{top} - \text{bottom}} = 5^{3-8}$ . The numerator exponent comes first!

Is my answer wrong if I get a negative exponent?

Not at all! Negative exponents are perfectly valid answers. They just represent fractions. $5^{-5}$ is the correct simplified form.

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