Simplify Powers: Finding 25^9 ÷ 25^2 Using Exponent Rules

Q: Why do I subtract the exponents when dividing?

Think of it as canceling! $ \frac{25^9}{25^2} $ means you have 9 factors of 25 on top and 2 factors of 25 on bottom. The bottom 2 factors cancel with 2 from the top, leaving you with $ 25^{9-2} = 25^7 $.

Q: What if I got 25^18 as my answer?

You probably multiplied the exponents instead of using division rules. Remember: multiplication of powers adds exponents, but division subtracts them. For $ \frac{25^9}{25^2} $, subtract: 9 - 2 = 7.

Q: Can I use this rule with different bases like 2^5 ÷ 3^2?

No! The bases must be exactly the same. You can only use $ a^m ÷ a^n = a^{m-n} $ when both terms have identical bases. Different bases cannot be simplified this way.

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

You still subtract! For example, $ \frac{25^2}{25^9} = 25^{2-9} = 25^{-7} $. The negative exponent means $ \frac{1}{25^7} $. Don't worry about negatives - just follow the subtraction rule!

Q: How can I remember this rule?

Remember MADSUB: Multiplication = Add exponents, Division = Subtract exponents. When you see division (÷ or fraction bar), think 'subtract the bottom from the top'!

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{25^9}{25^2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify the problem step by step.

00:10 We're going to use a formula for dividing powers.

00:14 When you divide a number A raised to the power of N, by the same base A raised to the power of M,

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

00:24 We'll practice using this formula in our exercise.

00:28 And that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{25^9}{25^2}=$

Step-by-step solution

To solve the expression $\frac{25^9}{25^2}$ , we will use the Power of a Quotient Rule for Exponents. According to this rule, when dividing like bases, we subtract the exponents.

$a^m \div a^n = a^{m-n}$

In the given expression, the base $25$ is the same for both the numerator and the denominator. Therefore, we can apply the rule as follows:

Identify the exponents: $m = 9$ and $n = 2$ .
Subtract the exponents: $9 - 2 = 7$ .
Write the result as a single power of the base: $25^7$ .

Thus, the expression $\frac{25^9}{25^2}$ simplifies to $25^7$ .

The solution to the question is: $25^7$

Final Answer

$25^7$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents
Technique: $\frac{25^9}{25^2}$ becomes $25^{9-2} = 25^7$
Check: Verify the exponent: 9 - 2 = 7, so answer is $25^7$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting when dividing
Don't add 9 + 2 = 11 to get $25^{11}$ ! Adding is for multiplication, not division. Always subtract exponents when dividing same bases: 9 - 2 = 7.

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?

Think of it as canceling! $\frac{25^9}{25^2}$ means you have 9 factors of 25 on top and 2 factors of 25 on bottom. The bottom 2 factors cancel with 2 from the top, leaving you with $25^{9-2} = 25^7$ .

What if I got 25^18 as my answer?

You probably multiplied the exponents instead of using division rules. Remember: multiplication of powers adds exponents, but division subtracts them. For $\frac{25^9}{25^2}$ , subtract: 9 - 2 = 7.

Can I use this rule with different bases like 2^5 ÷ 3^2?

No! The bases must be exactly the same. You can only use $a^m ÷ a^n = a^{m-n}$ when both terms have identical bases. Different bases cannot be simplified this way.

What happens if the bottom exponent is bigger than the top one?

You still subtract! For example, $\frac{25^2}{25^9} = 25^{2-9} = 25^{-7}$ . The negative exponent means $\frac{1}{25^7}$ . Don't worry about negatives - just follow the subtraction rule!

How can I remember this rule?

Remember MADSUB: Multiplication = Add exponents, Division = Subtract exponents. When you see division (÷ or fraction bar), think 'subtract the bottom from the top'!

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