Solve the Exponential Expression: a^x/a^y + a^2/a^x

Q: Why can't I add these two terms together?

You can only add terms with identical exponents! Since $ a^{x-y} $ and $ a^{2-x} $ have different exponents, they remain separate terms in your final answer.

Q: What if x equals y in the first term?

If x = y, then $ a^{x-y} = a^0 = 1 $! Remember that any number to the zero power equals 1, so your answer would be $ 1 + a^{2-x} $.

Q: Can I use this rule when the bases are different?

No! This division rule only works when the bases are exactly the same. If you have different bases like $ \frac{a^3}{b^2} $, you cannot simplify using exponent laws.

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

You still subtract! For example: $ \frac{a^2}{a^5} = a^{2-5} = a^{-3} $. Negative exponents are perfectly valid - they just mean $ \frac{1}{a^3} $.

Q: How do I remember which exponent to subtract from which?

Think of it as "top minus bottom": $ \frac{a^{\text{top}}}{a^{\text{bottom}}} = a^{\text{top - bottom}} $. The exponent in the numerator always comes first in the subtraction!

Exponential Division with Identical Bases

Solve the following equation:

$\frac{a^x}{a^y}+\frac{a^2}{a^x}=$

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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 exponents

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

00:15 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$\frac{a^x}{a^y}+\frac{a^2}{a^x}=$

Step-by-step solution

We will apply the law of exponents for division between terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$ Note: This law is only effective when the division is between terms with identical bases.

Let's return to the problem and apply the above law of exponents to each term in the sum separately:

$\frac{a^x}{a^y}+\frac{a^2}{a^x}=a^{x-y}+a^{2-x}$ Therefore the correct answer is A.

Final Answer

$a^{x-y}+a^{2-x}$

Key Points to Remember

Essential concepts to master this topic

Rule: For division with same bases: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Apply rule to each term separately: $\frac{a^x}{a^y} = a^{x-y}$ and $\frac{a^2}{a^x} = a^{2-x}$
Check: Final answer $a^{x-y} + a^{2-x}$ cannot be simplified further ✓

Common Mistakes

Avoid these frequent errors

Adding the exponents instead of subtracting
Don't add exponents like $\frac{a^x}{a^y} = a^{x+y}$ = completely wrong answer! This confuses division with multiplication. Always subtract the bottom exponent from the top exponent when dividing with identical bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I add these two terms together?

You can only add terms with identical exponents! Since $a^{x-y}$ and $a^{2-x}$ have different exponents, they remain separate terms in your final answer.

What if x equals y in the first term?

If x = y, then $a^{x-y} = a^0 = 1$ ! Remember that any number to the zero power equals 1, so your answer would be $1 + a^{2-x}$ .

Can I use this rule when the bases are different?

No! This division rule only works when the bases are exactly the same. If you have different bases like $\frac{a^3}{b^2}$ , you cannot simplify using exponent laws.

What happens if the bottom exponent is bigger than the top?

You still subtract! For example: $\frac{a^2}{a^5} = a^{2-5} = a^{-3}$ . Negative exponents are perfectly valid - they just mean $\frac{1}{a^3}$ .

How do I remember which exponent to subtract from which?

Think of it as "top minus bottom": $\frac{a^{\text{top}}}{a^{\text{bottom}}} = a^{\text{top - bottom}}$ . The exponent in the numerator always comes first in the subtraction!

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