Simplify the Expression: a^9 Divided by a^4 Using Power Rules

Q: Why do we subtract exponents when dividing?

Think of it as canceling out common factors! When you have $ \frac{a^9}{a^4} $, you're canceling 4 copies of 'a' from both top and bottom, leaving 5 copies on top: $ a^5 $.

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

You still subtract! For example, $ \frac{a^3}{a^7} = a^{3-7} = a^{-4} $. The negative exponent means one divided by that positive power.

Q: Do I always need to simplify a^{9-4} to a^5?

In most cases, yes! Teachers usually want the final simplified form $ a^5 $. However, $ a^{9-4} $ shows you correctly applied the rule.

Q: What if the bases are different letters?

The quotient rule only works with identical bases! You cannot simplify $ \frac{a^9}{b^4} $ because 'a' and 'b' are different variables.

Q: Can I use this rule with numbers too?

Absolutely! $ \frac{2^6}{2^3} = 2^{6-3} = 2^3 = 8 $. The quotient rule works with any identical base, whether it's a variable or a number.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the formula for dividing powers

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:06 equals the number (A) to the power of the difference of exponents (M-N)

00:08 We'll use this formula in our exercise

00:10 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\frac{a^9}{a^4}=$

2

Step-by-step solution

To solve this problem, we'll follow a systematic approach to simplify $\frac{a^9}{a^4}$ using exponent rules:

Step 1: Identify the expression.
We are given $\frac{a^9}{a^4}$ and asked to simplify it.
Step 2: Apply the Quotient Rule for Exponents.
According to the quotient rule, when dividing two expressions with the same base, we subtract the exponents. Given
$\frac{a^m}{a^n} = a^{m-n}$ ,
we apply this rule to our expression:

$a^{9-4}$

Step 3: Simplify the Expression.
We simplify the exponent by performing the subtraction:
$a^{9-4} = a^5$ .

Thus, the simplified form of the expression $\frac{a^9}{a^4}$ is $\mathbf{a^5}$ .

Now, let's match our simplified expression with the provided choices:

Choice 1: $a^{9-4}$ .
This reflects the correct expression before computation.
Choice 2: $a^{9+4}$ .
Incorrect, as it adds the exponents instead of subtracting.
Choice 3: $a^{9\times4}$ .
Incorrect, as it multiplies the exponents instead of subtracting.
Choice 4: $a^{\frac{9}{4}}$ .
Incorrect, as it divides the exponents instead of subtracting.

Hence, the correct choice is Choice 1: $a^{9-4}$ , which is the correct representation before simplification.

I am confident in the correctness of this solution.

3

Final Answer

$a^{9-4}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: $\frac{a^9}{a^4} = a^{9-4} = a^5$ by subtracting 4 from 9
Check: Expand to verify: $\frac{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a \cdot a} = a^5$ ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents like $a^{9+4} = a^{13}$ or multiply like $a^{9×4} = a^{36}$ ! This confuses division with multiplication or powers of powers. Always subtract exponents when dividing same bases: $\frac{a^9}{a^4} = a^{9-4}$ .

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 common factors! When you have $\frac{a^9}{a^4}$ , you're canceling 4 copies of 'a' from both top and bottom, leaving 5 copies on top: $a^5$ .

What if the bottom exponent is bigger than the top?

+

You still subtract! For example, $\frac{a^3}{a^7} = a^{3-7} = a^{-4}$ . The negative exponent means one divided by that positive power.

Do I always need to simplify a^{9-4} to a^5?

+

In most cases, yes! Teachers usually want the final simplified form $a^5$ . However, $a^{9-4}$ shows you correctly applied the rule.

What if the bases are different letters?

+

The quotient rule only works with identical bases! You cannot simplify $\frac{a^9}{b^4}$ because 'a' and 'b' are different variables.

Can I use this rule with numbers too?

+

Absolutely! $\frac{2^6}{2^3} = 2^{6-3} = 2^3 = 8$ . The quotient rule works with any identical base, whether it's a variable or a number.

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Simplify the Expression: a^9 Divided by a^4 Using Power Rules

Quotient Rule with Same Base Exponents

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