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

$(3\times4\times5)^4=$

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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