Simplify the Expression: a^7 Divided by a^3

Q: Why do we subtract exponents when dividing?

When you divide, you're canceling out matching factors! Think of $ \frac{a^7}{a^3} $ as $ \frac{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a} $. Three a's cancel out, leaving you with $ a^4 $.

Q: What if the bases are different letters?

This rule only works with identical bases! You can't simplify $ \frac{a^7}{b^3} $ because the bases are different. The bases must be exactly the same.

Q: Can the answer ever be a fraction?

Yes! If the bottom exponent is larger, like $ \frac{a^3}{a^7} = a^{3-7} = a^{-4} = \frac{1}{a^4} $. Negative exponents become fractions.

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

Multiplication = Add exponents: $ a^3 \cdot a^2 = a^5 $Division = Subtract exponents: $ \frac{a^5}{a^2} = a^3 $Think: multiplication builds up, division breaks down!

Q: What if there's a number in front of the variable?

Handle the numbers and variables separately! For $ \frac{6a^7}{2a^3} $, divide numbers: 6÷2=3, then subtract exponents: $ a^{7-3} = a^4 $. Final answer: $ 3a^4 $.

Exponent Division with Same Base

Simplify the following:

$\frac{a^7}{a^3}=$

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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 If we divide 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 exponents

00:15 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following:

$\frac{a^7}{a^3}=$

Step-by-step solution

Sincw 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:

$\frac{a^7}{a^3}=a^{7-3}=a^4$ Therefore, the correct answer is option C.

Final Answer

$a^4$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents
Technique: $\frac{a^7}{a^3} = a^{7-3} = a^4$ using subtraction
Check: Expand to verify: $\frac{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a} = a^4$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents like $a^7 \div a^3 = a^{10}$ ! Adding is for multiplication, not division. This gives the completely wrong answer. Always subtract the bottom exponent from the top exponent when dividing same bases.

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?

When you divide, you're canceling out matching factors! Think of $\frac{a^7}{a^3}$ as $\frac{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a}$ . Three a's cancel out, leaving you with $a^4$ .

What if the bases are different letters?

This rule only works with identical bases! You can't simplify $\frac{a^7}{b^3}$ because the bases are different. The bases must be exactly the same.

Can the answer ever be a fraction?

Yes! If the bottom exponent is larger, like $\frac{a^3}{a^7} = a^{3-7} = a^{-4} = \frac{1}{a^4}$ . Negative exponents become fractions.

How do I remember when to add vs subtract exponents?

Multiplication = Add exponents: $a^3 \cdot a^2 = a^5$
Division = Subtract exponents: $\frac{a^5}{a^2} = a^3$
Think: multiplication builds up, division breaks down!

What if there's a number in front of the variable?

Handle the numbers and variables separately! For $\frac{6a^7}{2a^3}$ , divide numbers: 6÷2=3, then subtract exponents: $a^{7-3} = a^4$ . Final answer: $3a^4$ .

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