Simplify (12×6)^20 ÷ (6×12)^4: Advanced Exponent Division

Q: Why are (12×6) and (6×12) considered the same base?

Because of the commutative property of multiplication! The order doesn't matter: 12×6 = 72 and 6×12 = 72. They're the same number, just written differently.

Q: What if the bases looked completely different?

If the bases were truly different (like 5^3 ÷ 2^3), you cannot use the quotient rule. You'd need to calculate each power separately first.

Q: How do I apply the quotient rule correctly?

When dividing powers with the same base: $ \frac{a^m}{a^n} = a^{m-n} $. Simply subtract the exponents: 20 - 4 = 16.

Q: Should I calculate 72^16 as my final answer?

Not unless specifically asked! The expression $ 72^{16} $ or $ (12×6)^{16} $ is the simplified form and perfectly acceptable as a final answer.

Q: What if I made an error and got 72^80?

You likely multiplied the exponents instead of subtracting! Remember: when dividing powers, you subtract exponents. When multiplying powers, you add them.

Exponent Division with Equivalent Bases

Insert the corresponding expression:

$\frac{\left(12\times6\right)^{20}}{\left(6\times12\right)^4}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 In multiplication, the order of factors doesn't matter

00:09 We'll use this formula in our exercise and reverse the order of factors

00:16 We'll use the formula for dividing powers

00:19 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:22 equals the number (A) to the power of the difference of exponents (M-N)

00:25 We'll use this formula in our exercise

00:35 Let's calculate the power

00:39 This is one solution method

00:45 Now let's switch the order of factors again and find a second solution

00:52 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:

$\frac{\left(12\times6\right)^{20}}{\left(6\times12\right)^4}=$

Step-by-step solution

To solve the expression $\frac{\left(12\times6\right)^{20}}{\left(6\times12\right)^4}$ , we will use the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$ .

First, let's simplify the expression inside the parentheses.

The numerator is: $(12 \times 6)^{20}$ and the denominator is: $(6 \times 12)^4$ .

Notice that $12 \times 6 = 72$ . Therefore, our expression simplifies to:

$\frac{72^{20}}{72^4}$

Applying the Power of a Quotient Rule, we have:

$72^{20-4} = 72^{16}$

Thus, the expression simplifies to $72^{16}$ .

The solution to the question is: $72^{16}$ . A'+C' are correct.

Final Answer

A'+C' are correct

Key Points to Remember

Essential concepts to master this topic

Recognition: Identify that 12×6 equals 6×12 by multiplication property
Technique: Apply quotient rule: $\frac{a^m}{a^n} = a^{m-n}$ gives $72^{16}$
Check: Verify 72^20 ÷ 72^4 = 72^16 using exponent subtraction ✓

Common Mistakes

Avoid these frequent errors

Treating different-looking expressions as different bases
Don't assume (12×6) and (6×12) are different bases = wrong calculations! These are identical values due to multiplication's commutative property. Always simplify expressions to recognize equivalent bases before applying exponent rules.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are (12×6) and (6×12) considered the same base?

Because of the commutative property of multiplication! The order doesn't matter: 12×6 = 72 and 6×12 = 72. They're the same number, just written differently.

What if the bases looked completely different?

If the bases were truly different (like 5^3 ÷ 2^3), you cannot use the quotient rule. You'd need to calculate each power separately first.

How do I apply the quotient rule correctly?

When dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$ . Simply subtract the exponents: 20 - 4 = 16.

Should I calculate 72^16 as my final answer?

Not unless specifically asked! The expression $72^{16}$ or $(12×6)^{16}$ is the simplified form and perfectly acceptable as a final answer.

What if I made an error and got 72^80?

You likely multiplied the exponents instead of subtracting! Remember: when dividing powers, you subtract exponents. When multiplying powers, you add them.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime