Simplify (4×5)⁸ ÷ (4×5)⁴: Applying Laws of Exponents

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{a^8}{a^4} $ means 8 copies of 'a' divided by 4 copies of 'a'. Four copies cancel out, leaving you with 8-4=4 copies, so $ a^4 $!

Q: What if the base is a multiplication like (4×5)?

Treat the entire expression $ (4\times5) $ as one single base. The quotient rule works the same way - just subtract the exponents normally!

Q: Can I simplify 4×5 to 20 first?

Yes, you can! $ (4\times5)^{8-4} = 20^4 $. Both forms are correct, but the question asks for the form with (4×5), so keep it that way.

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

You still subtract! For example: $ \frac{a^3}{a^7} = a^{3-7} = a^{-4} $. The negative exponent means one divided by that positive power.

Q: How can I remember when to add, subtract, or multiply exponents?

Multiplying: Add exponents → $ a^m \times a^n = a^{m+n} $Dividing: Subtract exponents → $ \frac{a^m}{a^n} = a^{m-n} $Power of power: Multiply exponents → $ (a^m)^n = a^{m \times n} $

Quotient Rule with Same Base Expressions

Insert the corresponding expression:

$\frac{\left(4\times5\right)^{8}}{\left(4\times5\right)^4}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's get started!

00:15 We'll use a neat formula for dividing powers.

00:19 Take any number, call it A, raised to the power N.

00:23 Divide it by the same number, A, to the power M.

00:27 It's equal to A raised to the power of M minus N.

00:32 Let's try this formula in our exercise.

00:35 And that's how you find the answer.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(4\times5\right)^{8}}{\left(4\times5\right)^4}=$

Step-by-step solution

We start with the given expression:
$\frac{\left(4\times5\right)^{8}}{\left(4\times5\right)^4}$

According to the power of a quotient rule for exponents, we can simplify an expression of the form $\frac{a^m}{a^n}$ as $a^{m-n}$ .
This rule states that when we divide two exponents with the same base, we subtract the exponents.

Applying this rule to our expression, we have:

Base: $4 \times 5$
Exponent in the numerator: $8$
Exponent in the denominator: $4$

Thus, we subtract the exponents in the quotient:

$(4\times5)^{8-4}$

Simplifying the exponent:

$(4\times5)^{4}$

Therefore, the expression simplifies to:
$(4\times5)^{8-4}$ .

The solution to the question is $\left(4\times5\right)^{8-4}$ .

Final Answer

$\left(4\times5\right)^{8-4}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Identify base $(4\times5)$ , then subtract: 8 - 4 = 4
Check: Verify $(4\times5)^4 = (4\times5)^{8-4}$ gives same result ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents (8+4=12) or multiply them (8×4=32) when dividing = completely wrong answer! Division with same bases always requires subtraction. Always subtract the bottom exponent from the top exponent.

Practice Quiz

Test your knowledge with interactive questions

Choose the corresponding expression:

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

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{a^8}{a^4}$ means 8 copies of 'a' divided by 4 copies of 'a'. Four copies cancel out, leaving you with 8-4=4 copies, so $a^4$ !

What if the base is a multiplication like (4×5)?

Treat the entire expression $(4\times5)$ as one single base. The quotient rule works the same way - just subtract the exponents normally!

Can I simplify 4×5 to 20 first?

Yes, you can! $(4\times5)^{8-4} = 20^4$ . Both forms are correct, but the question asks for the form with (4×5), so keep it that way.

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

You still subtract! For example: $\frac{a^3}{a^7} = a^{3-7} = a^{-4}$ . The negative exponent means one divided by that positive power.

How can I remember when to add, subtract, or multiply exponents?

Multiplying: Add exponents → $a^m \times a^n = a^{m+n}$
Dividing: Subtract exponents → $\frac{a^m}{a^n} = a^{m-n}$
Power of power: Multiply exponents → $(a^m)^n = a^{m \times n}$

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime