Simplify (2×3)⁶ ÷ (2×3)³: Power Division Problem

Q: Why do I subtract exponents when dividing but add them when multiplying?

Think of it this way: multiplication means adding more copies of the base, so exponents add up. Division means removing copies of the base, so exponents get smaller by subtraction!

Q: What if the base is more complicated like (2×3)?

The base can be any expression! Whether it's x, (2×3), or (a+b), the rule stays the same: $ \frac{base^m}{base^n} = base^{m-n} $.

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

You still subtract! If you get a negative exponent like $ a^{-2} $, that equals $ \frac{1}{a^2} $. The rule always works.

Q: How can I remember which operation to use?

Memory trick: Division and subtraction both make things smaller! So when you divide powers, subtract the exponents to make them smaller too.

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{\left(2\times3\right)^{6}}{\left(2\times3\right)^3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll 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 We'll 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:

$\frac{\left(2\times3\right)^{6}}{\left(2\times3\right)^3}=$

Step-by-step solution

Let's solve the given expression step by step by using the power of a quotient rule for exponents. The rule states that $\frac{a^n}{a^m} = a^{n-m}$ , where $a$ is any non-zero number, and $n$ and $m$ are integers.

Given the expression: $\frac{\left(2\times3\right)^{6}}{\left(2\times3\right)^3}$

First, apply the power of a quotient rule for exponents formula: $\frac{\left(2\times3\right)^{6}}{\left(2\times3\right)^3} = \left(2\times3\right)^{6-3}$ .
The exponent in the numerator is 6, and the exponent in the denominator is 3.
Subtract the exponent in the denominator from the exponent in the numerator: $6 - 3 = 3$ .
Thus, the expression simplifies to: $\left(2\times3\right)^3$ .

The solution to the question is: $\left(2\times3\right)^{6-3}$

Final Answer

$\left(2\times3\right)^{6-3}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Subtract denominator exponent from numerator exponent: 6 - 3 = 3
Check: Verify that $(2×3)^{6-3} = (2×3)^3 = 6^3 = 216$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of subtracting exponents during division
Don't add exponents like $(2×3)^{6+3} = (2×3)^9$ when dividing = wrong gigantic answer! Addition only applies to multiplication of same bases. 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 exponents when dividing but add them when multiplying?

Think of it this way: multiplication means adding more copies of the base, so exponents add up. Division means removing copies of the base, so exponents get smaller by subtraction!

What if the base is more complicated like (2×3)?

The base can be any expression! Whether it's x, (2×3), or (a+b), the rule stays the same: $\frac{base^m}{base^n} = base^{m-n}$ .

Can I simplify (2×3) to 6 first?

You could, but it's not necessary! The exponent rule works with any identical bases. Whether you write $(2×3)^{6-3}$ or $6^{6-3}$ , you get the same answer.

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

You still subtract! If you get a negative exponent like $a^{-2}$ , that equals $\frac{1}{a^2}$ . The rule always works.

How can I remember which operation to use?

Memory trick: Division and subtraction both make things smaller! So when you divide powers, subtract the exponents to make them smaller too.

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