Simplify the Expression: x⁹/x² Using Power Quotient Rule

Q: Why do we subtract exponents when dividing?

Think about what division means! $ \frac{x^9}{x^2} $ means we have 9 x's on top and 2 x's on bottom. We can cancel 2 x's from both, leaving us with 9 - 2 = 7 x's on top.

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

You still subtract! For example, $ \frac{x^2}{x^5} = x^{2-5} = x^{-3} $. A negative exponent means the variable moves to the denominator: $ \frac{1}{x^3} $.

Q: Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. You cannot simplify $ \frac{x^4}{y^2} $ using this rule because x and y are different bases.

Q: What happens if the exponents are equal?

When exponents are equal, like $ \frac{x^5}{x^5} $, you get $ x^{5-5} = x^0 = 1 $. Any non-zero number to the power of 0 equals 1!

Exponent Rules with Quotient Simplification

Insert the corresponding expression:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 First, let's simplify it.

00:09 We'll use a special formula for dividing powers.

00:13 If you have A to the power of N, divided by A to the power of M,

00:18 it equals A to the power of M minus N.

00:23 Let's apply this formula to our exercise.

00:26 And there you go, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

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

Step-by-step solution

To solve the expression $\frac{x^9}{x^2}$ , we will apply 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 our case, $a$ is $x$ , $m$ is 9, and $n$ is 2.

Now, apply the formula:

The expression can be rewritten as $x^{9-2}$ .
Calculate the exponent: $9 - 2 = 7$ .

Substitute back to get the simplified expression: $x^7$ .

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

Final Answer

$x^7$

Key Points to Remember

Essential concepts to master this topic

Power Quotient Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: For $\frac{x^9}{x^2}$ , calculate the new exponent: 9 - 2 = 7
Check: Verify by expanding: $\frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x} = x^7$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 9 + 2 = 11 to get $x^{11}$ ! Addition is for multiplication, not division. When dividing powers with the same base, always subtract the bottom exponent from the top exponent.

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 about what division means! $\frac{x^9}{x^2}$ means we have 9 x's on top and 2 x's on bottom. We can cancel 2 x's from both, leaving us with 9 - 2 = 7 x's on top.

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{x^2}{x^5} = x^{2-5} = x^{-3}$ . A negative exponent means the variable moves to the denominator: $\frac{1}{x^3}$ .

Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. You cannot simplify $\frac{x^4}{y^2}$ using this rule because x and y are different bases.

What happens if the exponents are equal?

When exponents are equal, like $\frac{x^5}{x^5}$ , you get $x^{5-5} = x^0 = 1$ . Any non-zero number to the power of 0 equals 1!

Do I need to expand everything to check my answer?

Not always! For simple cases like this, you can verify by thinking: "What times $x^2$ gives me $x^9$ ?" The answer is $x^7$ because $x^7 \cdot x^2 = x^9$ .

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