Simplify the Exponent Division: 13^17 ÷ 13^14

Q: Why do we subtract exponents when dividing?

Think of it as canceling out repeated multiplication! When you divide $ 13^{17} \div 13^{14} $, you're canceling 14 factors of 13, leaving you with $ 13^{17-14} = 13^3 $.

Q: What if the exponents are the same?

Great question! If you have $ \frac{13^{14}}{13^{14}} $, you get $ 13^{14-14} = 13^0 = 1 $. Any non-zero number to the power of 0 equals 1!

Q: Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. For $ \frac{13^5}{7^3} $, you cannot subtract exponents because 13 ≠ 7.

Q: What happens if the bottom exponent is bigger?

You still subtract! For example, $ \frac{13^2}{13^5} = 13^{2-5} = 13^{-3} $. The negative exponent means $ \frac{1}{13^3} $.

Q: How can I remember this rule?

Think: "Same base? Subtract!" Also remember that division is the opposite of multiplication, so while multiplication adds exponents, division subtracts them.

Quotient Rule with Same Base Exponents

Insert the corresponding expression:

$\frac{13^{17}}{13^{14}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's get started.

00:12 We'll use the rule for dividing powers.

00:15 If we have A to the power of N divided by A to the power of M,

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

00:23 We'll apply this rule in our exercise.

00:27 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{13^{17}}{13^{14}}=$

Step-by-step solution

To solve the expression $\frac{13^{17}}{13^{14}}$ , we use the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$ , where $a$ is a non-zero number, and $m$ and $n$ are integers.

In the given expression, $a = 13$ , $m = 17$ , and $n = 14$ . Applying the power of a quotient rule, we perform the following calculation:

Subtract the exponent in the denominator from the exponent in the numerator: $17 - 14 = 3$ .

This simplification leads us to:

$13^{17-14} = 13^3$

Therefore, the final simplified expression is $13^3$ .

Final Answer

$13^3$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract the exponents
Technique: $\frac{13^{17}}{13^{14}} = 13^{17-14} = 13^3$
Check: Verify by expanding: $13^3 = 13 \times 13 \times 13 = 2197$ ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents like 17 + 14 = 31 or multiply like 17 × 14! This confuses division with multiplication rules and gives massive wrong answers. 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 repeated multiplication! When you divide $13^{17} \div 13^{14}$ , you're canceling 14 factors of 13, leaving you with $13^{17-14} = 13^3$ .

What if the exponents are the same?

Great question! If you have $\frac{13^{14}}{13^{14}}$ , you get $13^{14-14} = 13^0 = 1$ . Any non-zero number to the power of 0 equals 1!

Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. For $\frac{13^5}{7^3}$ , you cannot subtract exponents because 13 ≠ 7.

What happens if the bottom exponent is bigger?

You still subtract! For example, $\frac{13^2}{13^5} = 13^{2-5} = 13^{-3}$ . The negative exponent means $\frac{1}{13^3}$ .

How can I remember this rule?

Think: "Same base? Subtract!" Also remember that division is the opposite of multiplication, so while multiplication adds exponents, division subtracts them.

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