Simplify the Exponential Expression: 9^15 ÷ 9^10

Q: Why do I subtract exponents when dividing?

Think of it like canceling! $ \frac{9^{15}}{9^{10}} $ means 15 nines multiplied together divided by 10 nines multiplied together. The 10 nines in the bottom cancel with 10 of the nines on top, leaving 5 nines: $ 9^5 $!

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

You still subtract! For example, $ \frac{9^{10}}{9^{15}} = 9^{10-15} = 9^{-5} $. The negative exponent means one divided by that positive power.

Q: Can I use this rule with different bases?

No! This rule only works when the bases are exactly the same. For $ \frac{9^{15}}{3^{10}} $, you'd need different methods since 9 ≠ 3.

Q: How do I remember which operation to use?

Use this memory trick: Same base division = Subtract exponents, Same base multiplication = Add exponents. Division and subtraction both 'take away'!

Q: Should I calculate 9^5 as my final answer?

Usually $ 9^5 $ is the simplified form teachers want. Only calculate the actual number (59,049) if specifically asked for the numerical value.

Exponential Division with Same Base

Insert the corresponding expression:

$\frac{9^{15}}{9^{10}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's start the simplification.

00:11 We're using the formula for dividing powers.

00:15 If you have a number A raised to the power of N,

00:18 and you divide it by the same base A to the power of M,

00:23 you get A to the power of M minus N.

00:27 Let's apply this formula in our exercise.

00:31 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:

$\frac{9^{15}}{9^{10}}=$

Step-by-step solution

To solve the expression $\frac{9^{15}}{9^{10}}$ , we will use the Power of a Quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ . This rule applies when both the numerator and the denominator have the same base.

In our problem, both the numerator and the denominator have the base 9, hence we can apply the rule:

Identify the exponents: The exponent in the numerator is 15, and the exponent in the denominator is 10.
Apply the Power of a Quotient rule by subtracting the exponent of the denominator from the exponent of the numerator:
$9^{15-10}$
Calculate the result of the subtraction:
$15 - 10 = 5$
Thus, the simplified form of the expression is:
$9^5$

The solution to the question is: $9^5$

Final Answer

$9^5$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{9^{15}}{9^{10}} = 9^{15-10} = 9^5$
Check: Verify by expanding: $9^5 = 59049$ equals original division ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents like 15 + 10 = 25 to get $9^{25}$ ! This confuses division with multiplication rules and gives a massive incorrect answer. 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 I subtract exponents when dividing?

Think of it like canceling! $\frac{9^{15}}{9^{10}}$ means 15 nines multiplied together divided by 10 nines multiplied together. The 10 nines in the bottom cancel with 10 of the nines on top, leaving 5 nines: $9^5$ !

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{9^{10}}{9^{15}} = 9^{10-15} = 9^{-5}$ . The negative exponent means one divided by that positive power.

Can I use this rule with different bases?

No! This rule only works when the bases are exactly the same. For $\frac{9^{15}}{3^{10}}$ , you'd need different methods since 9 ≠ 3.

How do I remember which operation to use?

Use this memory trick: Same base division = Subtract exponents, Same base multiplication = Add exponents. Division and subtraction both 'take away'!

Should I calculate 9^5 as my final answer?

Usually $9^5$ is the simplified form teachers want. Only calculate the actual number (59,049) if specifically asked for the numerical value.

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