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:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 Let's use the formula for dividing powers

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:06 equals the number (A) to the power of the difference of exponents (M-N)

00:08 Let's use this formula in our exercise

00:10 And this is the solution to the question

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

$112^0=\text{?}$

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 → $(a^m)^n = a^{m \times n}$

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