Simplify the Expression: a^a/a^b Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{a^a}{a^b} $ means how many times does $ a^b $ go into $ a^a $? Since $ a^a = a^b \cdot a^{a-b} $, the answer is $ a^{a-b} $!

Q: What if the bases are different?

You cannot use this rule! The division rule $ \frac{x^m}{x^n} = x^{m-n} $ only works when the bases are exactly the same. Different bases require other methods.

Q: Can the result have a negative exponent?

Yes! If $ a , then \( a^{a-b} $ will have a negative exponent. For example, $ \frac{x^2}{x^5} = x^{-3} $, which equals $ \frac{1}{x^3} $.

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

Use this memory trick: Multiplication = Addition ($ x^m \cdot x^n = x^{m+n} $) and Division = Subtraction ($ \frac{x^m}{x^n} = x^{m-n} $). Think "divide and subtract"!

Q: What happens if a = b in this problem?

If $ a = b $, then $ \frac{a^a}{a^b} = \frac{a^a}{a^a} = a^{a-a} = a^0 = 1 $ (assuming a ≠ 0). Any non-zero number to the power of 0 equals 1!

Exponent Division with Identical Bases

Simplify the following:

$\frac{a^a}{a^b}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem together.

00:09 When dividing powers with the same base, remember...

00:13 Subtract the exponents in the numerator and the denominator.

00:18 Let's apply this rule by subtracting the exponents.

00:22 Great job! That's how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following:

$\frac{a^a}{a^b}=$

Step-by-step solution

Since a division operation between two terms with identical bases is required, we will use the power property to divide terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$ Note that using this property is only possible when the division is performed between terms with identical bases.

We return to the problem and apply the mentioned power property:

$\frac{a^a}{a^b}=a^{a-b}$ Therefore, the correct answer is option D.

Final Answer

$a^{a-b}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^a}{a^b} = a^{a-b}$ by subtracting b from a
Check: Verify bases are identical before applying division rule ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents when dividing = $a^{a+b}$ which is wrong! Addition only works for multiplication of powers. Always subtract the bottom exponent from the top exponent when dividing.

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 this way: $\frac{a^a}{a^b}$ means how many times does $a^b$ go into $a^a$ ? Since $a^a = a^b \cdot a^{a-b}$ , the answer is $a^{a-b}$ !

What if the bases are different?

You cannot use this rule! The division rule $\frac{x^m}{x^n} = x^{m-n}$ only works when the bases are exactly the same. Different bases require other methods.

Can the result have a negative exponent?

Yes! If $a < b$ , then $a^{a-b}$ will have a negative exponent. For example, $\frac{x^2}{x^5} = x^{-3}$ , which equals $\frac{1}{x^3}$ .

How do I remember when to add vs subtract exponents?

Use this memory trick: Multiplication = Addition ( $x^m \cdot x^n = x^{m+n}$ ) and Division = Subtraction ( $\frac{x^m}{x^n} = x^{m-n}$ ). Think "divide and subtract"!

What happens if a = b in this problem?

If $a = b$ , then $\frac{a^a}{a^b} = \frac{a^a}{a^a} = a^{a-a} = a^0 = 1$ (assuming a ≠ 0). Any non-zero number to the power of 0 equals 1!

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