Simplify the Expression: a^9 Divided by a^x Step-by-Step

Q: Why do we subtract exponents when dividing?

Think of it as canceling out common factors! $ \frac{a^9}{a^x} $ means we have 9 copies of 'a' on top and x copies on bottom. After canceling x copies from both, we're left with $ a^{9-x} $.

Q: What happens if x is bigger than 9?

You still subtract normally! If x = 12, then $ \frac{a^9}{a^{12}} = a^{9-12} = a^{-3} $. The negative exponent means the answer goes in the denominator: $ \frac{1}{a^3} $.

Q: Can I use this rule with numbers instead of variables?

Absolutely! For example: $ \frac{3^8}{3^5} = 3^{8-5} = 3^3 = 27 $. The quotient rule works with any base, whether it's a number or variable.

Q: What if the bases are different?

The quotient rule only works with identical bases. You cannot simplify $ \frac{a^9}{b^x} $ using exponent rules because the bases (a and b) are different.

Q: How do I remember when to add vs subtract exponents?

Easy memory trick: Multiplication = Add, Division = Subtract. When you multiply $ a^m \cdot a^n $, add exponents. When you divide $ \frac{a^m}{a^n} $, subtract exponents!

Exponent Division with Variable Powers

Simplify the following expression:

$\frac{a^9}{a^x}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When dividing powers with equal bases

00:07 The power of the result equals the difference between the powers

00:11 We'll apply this formula to our exercise, and subtract the powers

00:14 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following expression:

$\frac{a^9}{a^x}$

Step-by-step solution

In the question there is a fraction that has terms with identical bases in its numerator and denominator. Therefore, so we can use the distributive property of division to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$

We apply the previously distributive property to the problem:

$\frac{a^9}{a^x}=a^{9-x}$

Therefore, the correct answer is (c).

Final Answer

$a^{9-x}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^9}{a^x} = a^{9-x}$ using quotient rule
Check: Multiply result by denominator: $a^{9-x} \cdot a^x = a^9$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't use $\frac{a^9}{a^x} = a^{9+x}$ = wrong operation! Addition rule applies to multiplication, not division. Always subtract the bottom exponent from the top exponent when dividing.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it as canceling out common factors! $\frac{a^9}{a^x}$ means we have 9 copies of 'a' on top and x copies on bottom. After canceling x copies from both, we're left with $a^{9-x}$ .

What happens if x is bigger than 9?

You still subtract normally! If x = 12, then $\frac{a^9}{a^{12}} = a^{9-12} = a^{-3}$ . The negative exponent means the answer goes in the denominator: $\frac{1}{a^3}$ .

Can I use this rule with numbers instead of variables?

Absolutely! For example: $\frac{3^8}{3^5} = 3^{8-5} = 3^3 = 27$ . The quotient rule works with any base, whether it's a number or variable.

What if the bases are different?

The quotient rule only works with identical bases. You cannot simplify $\frac{a^9}{b^x}$ using exponent rules because the bases (a and b) are different.

How do I remember when to add vs subtract exponents?

Easy memory trick: Multiplication = Add, Division = Subtract. When you multiply $a^m \cdot a^n$ , add exponents. When you divide $\frac{a^m}{a^n}$ , subtract exponents!

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