Simplify (a×b)^15 Divided by (a×b)^3: Power Division Problem

Q: Why do I subtract exponents when dividing?

Think of it as canceling out! When you have $ \frac{(ab)^{15}}{(ab)^3} $, you're canceling 3 copies of (ab) from both top and bottom, leaving 15 - 3 = 12 copies on top.

Q: What if the base isn't exactly the same?

The bases must be identical to use this rule. $ (a \times b) $ and $ (b \times a) $ are the same base because multiplication is commutative, but $ a^2b $ is different!

Q: Can the answer have a negative exponent?

Yes! If the bottom exponent is bigger, like $ \frac{x^3}{x^7} = x^{-4} $, you get a negative exponent. This equals $ \frac{1}{x^4} $.

Q: How do I remember which operation to use?

Division = Subtraction, Multiplication = Addition. Think: "Dividing powers? Subtract the powers!" It's the opposite of what you do when multiplying powers.

Q: What if one of the exponents is zero?

Remember that any base to the power of 0 equals 1. So $ \frac{x^5}{x^0} = \frac{x^5}{1} = x^5 $, which matches our rule: 5 - 0 = 5!

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{\left(a\times b\right)^{15}}{\left(a\times b\right)^3}=$

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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)

00:07 equals the number (A) to the power of the difference of exponents (M-N)

00:10 We'll use this formula in our exercise

00:13 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:

$\frac{\left(a\times b\right)^{15}}{\left(a\times b\right)^3}=$

Step-by-step solution

The given expression is:
$\frac{\left(a\times b\right)^{15}}{\left(a\times b\right)^3}$

We need to apply the division rule for exponents, which states that:
$\frac{x^m}{x^n} = x^{m-n}$

Using this rule, we can simplify the expression by subtracting the exponent of the denominator from the exponent of the numerator:
$\frac{\left(a\times b\right)^{15}}{\left(a\times b\right)^3} = \left(a\times b\right)^{15-3}$

Subtracting the exponents, we have:
$\left(a\times b\right)^{12}$

Therefore, the simplified expression is:
$\left(a\times b\right)^{12}$

The solution to the question is:
$\left(a\times b\right)^{12}$

Final Answer

$\left(a\times b\right)^{12}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{(ab)^{15}}{(ab)^3} = (ab)^{15-3} = (ab)^{12}$
Check: Count how many times the base cancels out: 15 - 3 = 12 ✓

Common Mistakes

Avoid these frequent errors

Adding instead of subtracting exponents
Don't add exponents like 15 + 3 = 18 when dividing! This gives $(ab)^{18}$ which is completely wrong. Division means subtraction, multiplication means addition. 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 I subtract exponents when dividing?

Think of it as canceling out! When you have $\frac{(ab)^{15}}{(ab)^3}$ , you're canceling 3 copies of (ab) from both top and bottom, leaving 15 - 3 = 12 copies on top.

What if the base isn't exactly the same?

The bases must be identical to use this rule. $(a \times b)$ and $(b \times a)$ are the same base because multiplication is commutative, but $a^2b$ is different!

Can the answer have a negative exponent?

Yes! If the bottom exponent is bigger, like $\frac{x^3}{x^7} = x^{-4}$ , you get a negative exponent. This equals $\frac{1}{x^4}$ .

How do I remember which operation to use?

Division = Subtraction, Multiplication = Addition. Think: "Dividing powers? Subtract the powers!" It's the opposite of what you do when multiplying powers.

What if one of the exponents is zero?

Remember that any base to the power of 0 equals 1. So $\frac{x^5}{x^0} = \frac{x^5}{1} = x^5$ , which matches our rule: 5 - 0 = 5!

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