Simplify (8×7)^15 Divided by (8×7)^3: Power Division Problem

Q: Why do we subtract exponents when dividing, but add when multiplying?

Think of it this way: multiplication combines repeated factors, so exponents add. Division removes factors from the numerator, so we subtract! $ \frac{a^5}{a^2} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a} = a^3 $

Q: What if the base is a product like (8×7)?

It doesn't matter! Treat $ (8\times7) $ as one complete base. The exponent rules work exactly the same: $ \frac{(8\times7)^{15}}{(8\times7)^3} = (8\times7)^{15-3} $

Q: Can I simplify (8×7) first before applying the exponent rule?

Yes, but it's not necessary! You could calculate 8×7 = 56 first, but the exponent rule works either way. Keep it as $ (8\times7)^{12} $ or simplify to $ 56^{12} $.

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

You still subtract! If you have $ \frac{a^3}{a^7} = a^{3-7} = a^{-4} $, you get a negative exponent, which equals $ \frac{1}{a^4} $.

Q: How can I remember this rule?

Use this memory trick: "Same base division = Subtract the station!" The exponents are like station numbers - when dividing, you subtract to find how many stations apart they are.

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{\left(8\times7\right)^{15}}{\left(8\times7\right)^3}=$

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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

00:12 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:

$\frac{\left(8\times7\right)^{15}}{\left(8\times7\right)^3}=$

Step-by-step solution

We are given the expression: $\frac{\left(8\times7\right)^{15}}{\left(8\times7\right)^3}$

To solve this, we can use the Power of a Quotient Rule for Exponents. This rule states that for any non-zero numbers $a$ and $b$ , and any integers $m$ and $n$ , the expression:

$\frac{a^m}{a^n} = a^{m-n}$

can be simplified by subtracting the exponent in the denominator from the exponent in the numerator.

Using the Power of a Quotient Rule, let's apply it to our expression:

Given: $\frac{\left(8\times7\right)^{15}}{\left(8\times7\right)^3}$

According to the rule: $\left(8\times7\right)^{15-3}$

So, the simplified expression is: $\left(8\times7\right)^{12}$

Thus, the correct simplified expression is: $\left(8\times7\right)^{15-3}$

Final Answer

$\left(8\times7\right)^{15-3}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract the exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ , so 15 - 3 = 12
Check: Verify by expanding: same base appears in numerator and denominator ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying the exponents instead of subtracting
Don't add exponents (15 + 3 = 18) or multiply them (15 × 3 = 45) when dividing powers = completely wrong operations! Division of same bases requires subtraction of exponents. 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, but add when multiplying?

Think of it this way: multiplication combines repeated factors, so exponents add. Division removes factors from the numerator, so we subtract! $\frac{a^5}{a^2} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a} = a^3$

What if the base is a product like (8×7)?

It doesn't matter! Treat $(8\times7)$ as one complete base. The exponent rules work exactly the same: $\frac{(8\times7)^{15}}{(8\times7)^3} = (8\times7)^{15-3}$

Can I simplify (8×7) first before applying the exponent rule?

Yes, but it's not necessary! You could calculate 8×7 = 56 first, but the exponent rule works either way. Keep it as $(8\times7)^{12}$ or simplify to $56^{12}$ .

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

You still subtract! If you have $\frac{a^3}{a^7} = a^{3-7} = a^{-4}$ , you get a negative exponent, which equals $\frac{1}{a^4}$ .

How can I remember this rule?

Use this memory trick: "Same base division = Subtract the station!" The exponents are like station numbers - when dividing, you subtract to find how many stations apart they are.

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