Simplify (9×7)^9 ÷ (9×7)^4: Exponential Division Problem

Q: Why do we subtract exponents when dividing?

Think of it as cancelling! When you divide $ \frac{a^9}{a^4} $, you can cancel out 4 of the 9 factors, leaving you with $ a^5 $. Subtraction counts what's left over.

Q: What if the base is more complicated like (9×7)?

It doesn't matter how complex the base is! The rule $ \frac{a^m}{a^n} = a^{m-n} $ works for any base, whether it's a number, variable, or expression like (9×7).

Q: Do I need to calculate 9×7 = 63 first?

No! Keep the base as (9×7) throughout your work. The division rule applies regardless of whether you simplify the base or not. Your final answer is $ (9×7)^5 $.

Q: How can I remember when to add vs subtract exponents?

Use this memory trick: Multiplication = Addition of exponents, Division = Subtraction of exponents. Think 'M&A' and 'D&S'!

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

You still subtract! For example, $ \frac{a^4}{a^7} = a^{4-7} = a^{-3} $. The negative exponent means one over the positive power: $ \frac{1}{a^3} $.

Exponential Division with Same Base

Insert the corresponding expression:

$\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 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 Let's use this formula in our exercise

00:12 Let's calculate the power

00:14 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:

$\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4}=$

Step-by-step solution

Let's solve the given expression: $\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4}$ .

We need to use the Power of a Quotient Rule for exponents. The rule states that:

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

In our expression, $a = 9 \times 7$ , $m = 9$ , and $n = 4$ .

Applying the rule, we have:

$\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4} = \left(9\times7\right)^{9-4}$

Perform the subtraction in the exponent:

$9 - 4 = 5$

So the expression simplifies to:

$\left(9\times7\right)^5$

Therefore, the expression $\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4}$ simplifies to $\left(9\times7\right)^5$ , which is indeed the correct answer.

Final Answer

$\left(9\times7\right)^5$

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 $\frac{(9×7)^9}{(9×7)^4} = (9×7)^{9-4}$
Check: Verify by multiplying: $(9×7)^5 × (9×7)^4 = (9×7)^9$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents 9 + 4 = 13! This gives $(9×7)^{13}$ which is wrong because adding exponents only applies to multiplication. Always subtract exponents when dividing: $\frac{a^m}{a^n} = a^{m-n}$ .

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 as cancelling! When you divide $\frac{a^9}{a^4}$ , you can cancel out 4 of the 9 factors, leaving you with $a^5$ . Subtraction counts what's left over.

What if the base is more complicated like (9×7)?

It doesn't matter how complex the base is! The rule $\frac{a^m}{a^n} = a^{m-n}$ works for any base, whether it's a number, variable, or expression like (9×7).

Do I need to calculate 9×7 = 63 first?

No! Keep the base as (9×7) throughout your work. The division rule applies regardless of whether you simplify the base or not. Your final answer is $(9×7)^5$ .

How can I remember when to add vs subtract exponents?

Use this memory trick: Multiplication = Addition of exponents, Division = Subtraction of exponents. Think 'M&A' and 'D&S'!

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

You still subtract! For example, $\frac{a^4}{a^7} = a^{4-7} = a^{-3}$ . The negative exponent means one over the positive power: $\frac{1}{a^3}$ .

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