Simplify (6×3)^7 ÷ (3×6)^2: Advanced Exponent Division

Q: Why do 6×3 and 3×6 give the same result?

Because multiplication is commutative! This means the order doesn't matter: 6×3 = 3×6 = 18. Once you recognize this, the problem becomes much simpler.

Q: Can I subtract exponents even when the expressions look different?

Only when the bases are actually equal! Since (6×3) = (3×6) = 18, we can use the rule $ \frac{a^m}{a^n} = a^{m-n} $ to get $ 18^{7-2} = 18^5 $.

Q: What if the answer choices show the expression differently?

Look for equivalent forms! Since 6×3 = 3×6 = 18, expressions like $ (6×3)^5 $, $ (3×6)^5 $, and $ 18^5 $ are all correct.

Q: How do I know which exponent rule to use?

When dividing powers with the same base, use $ \frac{a^m}{a^n} = a^{m-n} $. The key is first recognizing that the bases are equivalent!

Exponent Division with Equivalent Bases

Insert the corresponding expression:

$\frac{\left(6\times3\right)^7}{\left(3\times6\right)^2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Let's begin by remembering something simple.

00:17 In multiplication, the order of numbers doesn't change the result.

00:22 We'll practice this by reversing the order of our numbers.

00:26 Next, we apply our formula for dividing powers.

00:31 Any base, like A, raised to the power of N divided by the same base A to the power of M.

00:37 This equals the base A raised to the power of M minus N.

00:42 Let's apply this formula in our exercise.

00:46 Now, let's calculate the power together.

00:49 And we'll switch the order back, using our math skill.

00:54 Great job! That's how we solve this question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(6\times3\right)^7}{\left(3\times6\right)^2}=$

Step-by-step solution

Let's solve the expression step-by-step using the power of a quotient rule for exponents.

The provided expression is:

$\frac{\left(6\times3\right)^7}{\left(3\times6\right)^2}$

The power of a quotient rule states:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

First, simplify the terms inside the parentheses:

The numerator: $6 \times 3 = 18$
The denominator: $3 \times 6 = 18$

Thus, the expression simplifies to:

$\frac{18^7}{18^2}$

We can simplify this further using the rule of dividing powers with the same base:

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

Apply the rule:

$\frac{18^7}{18^2} = 18^{7-2} = 18^5$

The solution to the question is:

Final Answer

A+B are correct

Key Points to Remember

Essential concepts to master this topic

Recognition: Identify that 6×3 and 3×6 both equal 18
Technique: Apply $\frac{a^m}{a^n} = a^{m-n}$ to get $18^{7-2} = 18^5$
Check: Verify that $(6×3)^{7-2} = (18)^5$ matches our answer ✓

Common Mistakes

Avoid these frequent errors

Not recognizing that 6×3 equals 3×6
Don't treat (6×3)^7 and (3×6)^2 as different bases = makes the problem seem impossible! Multiplication is commutative, so both equal 18. Always simplify expressions inside parentheses first to reveal equivalent bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do 6×3 and 3×6 give the same result?

Because multiplication is commutative! This means the order doesn't matter: 6×3 = 3×6 = 18. Once you recognize this, the problem becomes much simpler.

Can I subtract exponents even when the expressions look different?

Only when the bases are actually equal! Since (6×3) = (3×6) = 18, we can use the rule $\frac{a^m}{a^n} = a^{m-n}$ to get $18^{7-2} = 18^5$ .

What if the answer choices show the expression differently?

Look for equivalent forms! Since 6×3 = 3×6 = 18, expressions like $(6×3)^5$ , $(3×6)^5$ , and $18^5$ are all correct.

How do I know which exponent rule to use?

When dividing powers with the same base, use $\frac{a^m}{a^n} = a^{m-n}$ . The key is first recognizing that the bases are equivalent!

Should I calculate 18^5 to get the final number?

Usually no! Unless specifically asked, leave your answer in exponential form like $18^5$ or $(6×3)^5$ since these are much easier to work with.

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