Simplify (9×5)¹² ÷ (5×9)⁶: Power Division Challenge

Q: Why can I treat (9×5) and (5×9) as the same base?

Because multiplication is commutative! This means $ 9×5 = 5×9 = 45 $. Both expressions equal exactly the same value, so they're the same base.

Q: What if the bases looked different but were actually the same?

Always simplify first! For example, $ 2^3 $ and $ 8 $ are the same base since $ 2^3 = 8 $. Look for equivalent expressions.

Q: Why do I subtract exponents when dividing?

Think of it this way: $ \frac{a^5}{a^2} = \frac{a·a·a·a·a}{a·a} $. You can cancel two a's from top and bottom, leaving $ a^3 $, which is $ a^{5-2} $!

Q: What would happen if I added the exponents instead?

You'd get $ (9×5)^{18} $ instead of $ (9×5)^6 $ - a huge difference! Adding exponents is for multiplication, not division.

Q: How can I remember the rule?

Remember: Division = Subtraction for exponents. When you divide, you're reducing the power, so subtract. When you multiply, you add exponents.

Exponent Division with Identical Bases

Insert the corresponding expression:

$\frac{\left(9\times5\right)^{12}}{\left(5\times9\right)^6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's simplify the exercise.

00:15 In multiplication, the order of numbers doesn't matter.

00:19 We'll use this concept and switch the order of numbers in our exercise.

00:27 Now, let's look at dividing powers.

00:30 If you have a number, A, to the power of N, divided by A to the power of M,

00:36 it equals A to the power of M minus N.

00:41 We'll apply this formula in our exercise.

00:45 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(9\times5\right)^{12}}{\left(5\times9\right)^6}=$

Step-by-step solution

We begin by analyzing the given expression: $\frac{\left(9\times5\right)^{12}}{\left(5\times9\right)^6}$ . Using the property of exponents known as the Power of a Quotient Rule, we can rewrite this expression.
This rule states that $\frac{a^m}{a^n} = a^{m-n}$ . Here, both the numerator and the denominator have the same base, $9\times5$ or equivalently $5\times9$ , therefore we can apply this rule.

Let's apply the Power of a Quotient Rule:

Identify the base, which is $9\times5$ .
Subtract the exponent in the denominator from the exponent in the numerator: $12 - 6$ .

Thus, the expression simplifies to $\left(9\times5\right)^{12-6}$ .

The solution to the question is: $\left(9\times5\right)^{12-6}$ .

Final Answer

$\left(9\times5\right)^{12-6}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ so 12 - 6 = 6
Check: Both $(9×5)$ and $(5×9)$ equal 45, confirming same base ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents (12+6) or multiply them (12×6) when dividing = completely wrong answer! Division of powers requires subtraction, not addition or multiplication. Always subtract the bottom exponent from the top exponent.

Practice Quiz

Test your knowledge with interactive questions

$5^4\times25=$

FAQ

Everything you need to know about this question

Why can I treat (9×5) and (5×9) as the same base?

Because multiplication is commutative! This means $9×5 = 5×9 = 45$ . Both expressions equal exactly the same value, so they're the same base.

What if the bases looked different but were actually the same?

Always simplify first! For example, $2^3$ and $8$ are the same base since $2^3 = 8$ . Look for equivalent expressions.

Why do I subtract exponents when dividing?

Think of it this way: $\frac{a^5}{a^2} = \frac{a·a·a·a·a}{a·a}$ . You can cancel two a's from top and bottom, leaving $a^3$ , which is $a^{5-2}$ !

What would happen if I added the exponents instead?

You'd get $(9×5)^{18}$ instead of $(9×5)^6$ - a huge difference! Adding exponents is for multiplication, not division.

How can I remember the rule?

Remember: Division = Subtraction for exponents. When you divide, you're reducing the power, so subtract. When you multiply, you add exponents.

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