Simplify a³/a¹: Step-by-Step Exponent Division

Q: Why do I subtract the exponents when dividing?

Think about what division means! $ \frac{a^3}{a^1} = \frac{a \cdot a \cdot a}{a} $. You can cancel out one 'a' from top and bottom, leaving $ a \cdot a = a^2 $.

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

You still subtract! For example, $ \frac{a^2}{a^5} = a^{2-5} = a^{-3} $. The negative exponent means one over that positive power.

Q: Do the bases have to be exactly the same letter?

Yes! This rule only works when the bases are identical. You can't use it for $ \frac{a^3}{b^1} $ because 'a' and 'b' are different variables.

Q: What happens if the exponent in the denominator is 1?

An exponent of 1 is often invisible but still there! $ a^1 = a $, so $ \frac{a^3}{a} $ is the same as $ \frac{a^3}{a^1} $.

Q: How can I remember this rule?

Think: "Division = Subtraction" for exponents! Just like $ \frac{10^5}{10^2} = 10^{5-2} = 10^3 $, you're removing powers from the top.

Exponent Division with Same Base

Simplify the following:

$\frac{a^3}{a^1}=$

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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:

$\frac{a^3}{a^1}=$

Step-by-step solution

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

$\frac{a^3}{a^1}=a^{3-1}=a^2$ Therefore, the correct answer is option A.

Final Answer

$a^2$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^3}{a^1} = a^{3-1} = a^2$
Check: Expand to verify: $\frac{a \cdot a \cdot a}{a} = a \cdot a = a^2$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 3 + 1 = 4 to get $a^4$ ! Adding is for multiplication, not division, which gives the wrong answer. 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 I subtract the exponents when dividing?

Think about what division means! $\frac{a^3}{a^1} = \frac{a \cdot a \cdot a}{a}$ . You can cancel out one 'a' from top and bottom, leaving $a \cdot a = a^2$ .

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

You still subtract! For example, $\frac{a^2}{a^5} = a^{2-5} = a^{-3}$ . The negative exponent means one over that positive power.

Do the bases have to be exactly the same letter?

Yes! This rule only works when the bases are identical. You can't use it for $\frac{a^3}{b^1}$ because 'a' and 'b' are different variables.

What happens if the exponent in the denominator is 1?

An exponent of 1 is often invisible but still there! $a^1 = a$ , so $\frac{a^3}{a}$ is the same as $\frac{a^3}{a^1}$ .

How can I remember this rule?

Think: "Division = Subtraction" for exponents! Just like $\frac{10^5}{10^2} = 10^{5-2} = 10^3$ , you're removing powers from the top.

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