Simplify the Algebraic Fraction: 4a⁵/2a³ Using Exponent Rules

Q: Why do I subtract the 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} $. When you cancel the three a's on bottom with three a's on top, you're left with $ a^2 $!

Q: What about the numbers 4 and 2 - do I use exponent rules on them too?

No! The numbers 4 and 2 are just coefficients. Treat them like regular division: $ \frac{4}{2} = 2 $. Only use exponent rules on the variables with powers.

Q: Can I simplify this problem differently?

Yes! You could also think of it as: $ \frac{4a^5}{2a^3} = \frac{4a^5}{2a^3} \cdot \frac{1/a^3}{1/a^3} = \frac{4a^2}{2} = 2a^2 $. Both methods give the same answer.

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} = \frac{1}{a^2} $. Negative exponents mean the variable goes to the denominator.

Q: How do I check my answer is correct?

Substitute a simple value like a = 2. Original: $ \frac{4(2)^5}{2(2)^3} = \frac{4 \cdot 32}{2 \cdot 8} = \frac{128}{16} = 8 $. Your answer: $ 2(2)^2 = 2 \cdot 4 = 8 $ ✓

Exponent Division with Identical Bases

Solve the exercise:

$\frac{4a^5}{2a^3}=$

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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

00:15 We'll apply this formula to our exercise, and subtract the powers

00:23 Let's calculate 4 divided by 2

00:28 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the exercise:

$\frac{4a^5}{2a^3}=$

Step-by-step solution

Due to the fact that the numerator and the denominator of the fraction have terms with identical bases, we will begin by applying the law of exponents for the division of terms with identical bases:

$\frac{b^m}{b^n}=b^{m-n}$ We begin by applying the formula to the given problem:

$\frac{4a^5}{2a^3}=2\cdot a^{5-3}=2\cdot a^2$ In the first step we simplify the numerical part of the fraction. This is simple to do and makes it easier to work with the said fraction.

$\frac{4a^5}{2a^3}=\frac{4}{2}\cdot\frac{a^5}{a^3}=2\cdot a^{5-3}=\ldots$ We obtain the following answer:

$2a^2$

Therefore, the correct answer is option A.

Final Answer

$2a^2$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: Separate coefficients: $\frac{4a^5}{2a^3} = \frac{4}{2} \cdot \frac{a^5}{a^3} = 2 \cdot a^{5-3}$
Check: Expand answer: $2a^2 = 2 \cdot a \cdot a$ matches simplified form ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents when dividing: $\frac{a^5}{a^3} \neq a^{5+3} = a^8$ ! This gives the wrong answer because you're multiplying instead of dividing. 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 I subtract the exponents when dividing?

Think of it this way: $\frac{a^5}{a^3} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a}$ . When you cancel the three a's on bottom with three a's on top, you're left with $a^2$ !

What about the numbers 4 and 2 - do I use exponent rules on them too?

No! The numbers 4 and 2 are just coefficients. Treat them like regular division: $\frac{4}{2} = 2$ . Only use exponent rules on the variables with powers.

Can I simplify this problem differently?

Yes! You could also think of it as: $\frac{4a^5}{2a^3} = \frac{4a^5}{2a^3} \cdot \frac{1/a^3}{1/a^3} = \frac{4a^2}{2} = 2a^2$ . Both methods give the same answer.

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

You still subtract! For example: $\frac{a^3}{a^5} = a^{3-5} = a^{-2} = \frac{1}{a^2}$ . Negative exponents mean the variable goes to the denominator.

How do I check my answer is correct?

Substitute a simple value like a = 2. Original: $\frac{4(2)^5}{2(2)^3} = \frac{4 \cdot 32}{2 \cdot 8} = \frac{128}{16} = 8$ . Your answer: $2(2)^2 = 2 \cdot 4 = 8$ ✓

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