Simplify a⁵/a³: Exponent Division Practice Problem

Q: Why do I subtract exponents when dividing?

Think of it this way: $ \frac{a^5}{a^3} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a} $. The three a's in the denominator cancel with three in the numerator, leaving two a's or $ a^2 $!

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

You still subtract! For example, $ \frac{a^3}{a^5} = a^{3-5} = a^{-2} $. The negative exponent means the answer goes in the denominator: $ \frac{1}{a^2} $.

Q: Can I use this rule with numbers too?

Absolutely! For example: $ \frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4 $. The rule works for any base, whether it's a variable or a number.

Q: What if the bases are different?

The exponent division rule only works when the bases are identical. For $ \frac{a^5}{b^3} $, you cannot subtract exponents because the bases are different.

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

Multiplication: Add exponents → $ a^m \cdot a^n = a^{m+n} $Division: Subtract exponents → $ \frac{a^m}{a^n} = a^{m-n} $Think: multiply = more = add, divide = less = subtract!

Exponent Division with Same Base Variables

Simplify the following:

$\frac{a^5}{a^3}=$

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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 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:

$\frac{a^5}{a^3}=$

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 carried out between terms with identical bases.

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

$\frac{a^5}{a^3}=a^{5-3}=a^2$ Therefore, the correct answer is option A.

Final Answer

$a^2$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When dividing same bases, subtract the exponents
Technique: $\frac{a^5}{a^3} = a^{5-3} = a^2$
Check: Multiply result by denominator: $a^2 \cdot a^3 = a^5$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents 5 + 3 = 8 to get $a^8$ ! Addition is for multiplication of same bases, not division. Always subtract exponents when dividing: $\frac{a^m}{a^n} = a^{m-n}$ .

Practice Quiz

Test your knowledge with interactive questions

$5^4\times25=$

FAQ

Everything you need to know about this question

Why do I subtract exponents when dividing?

Think of it this way: $\frac{a^5}{a^3} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a}$ . The three a's in the denominator cancel with three in the numerator, leaving two a's or $a^2$ !

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

You still subtract! For example, $\frac{a^3}{a^5} = a^{3-5} = a^{-2}$ . The negative exponent means the answer goes in the denominator: $\frac{1}{a^2}$ .

Can I use this rule with numbers too?

Absolutely! For example: $\frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4$ . The rule works for any base, whether it's a variable or a number.

What if the bases are different?

The exponent division rule only works when the bases are identical. For $\frac{a^5}{b^3}$ , you cannot subtract exponents because the bases are different.

How can I remember when to add vs. subtract exponents?

Multiplication: Add exponents → $a^m \cdot a^n = a^{m+n}$
Division: Subtract exponents → $\frac{a^m}{a^n} = a^{m-n}$

Think: multiply = more = add, divide = less = subtract!

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