Expand (2·6·3)^9: Step-by-Step Product Power Calculation

Q: Why do I need to apply the exponent to every factor?

The power of a product rule states that when you raise a product to a power, you must raise each factor to that power. This is like distributing multiplication - the exponent affects everything inside the parentheses!

Q: What if I forget to apply the exponent to one factor?

Your answer will be completely wrong! For example, $ 2 \times 6^9 \times 3^9 $ gives a much smaller result than the correct $ 2^9 \times 6^9 \times 3^9 $ because you're missing $ 2^8 $ as a factor.

Q: Can I simplify the expression inside first?

Yes! You could calculate $ 2 \cdot 6 \cdot 3 = 36 $ first, then find $ 36^9 $. Both methods give the same answer, but expanding first shows the structure more clearly.

Q: How do I know if all the answer choices are wrong?

Compare each choice to your correct expansion. If none match $ 2^9 \times 6^9 \times 3^9 $, then "None of the answers are correct" is the right choice!

Q: Does the order of factors matter in my answer?

No! Multiplication is commutative, so $ 2^9 \times 6^9 \times 3^9 $, $ 6^9 \times 2^9 \times 3^9 $, and other arrangements are all equivalent.

Power of Products with Multiple Factors

Choose the expression that corresponds to the following:

$\left(2\cdot6\cdot3\right)^9=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When we are presented with a multiplication operation where all the factors have the same exponent (N)

00:08 Each factor can be raised to the power (N)

00:12 We'll apply this formula to our exercise

00:23 Let's proceed to review the incorrect options

00:29 This option is incorrect due to the fact that factor 2 is not raised to the power

00:40 The same thing with this option but with factor 3

00:46 And this one with factor 6

00:53 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(2\cdot6\cdot3\right)^9=$

Step-by-step solution

To solve this problem, let's expand the expression $(2 \cdot 6 \cdot 3)^9$ .

Step 1: Apply the power of a product rule: $(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$ .

Step 2: Identify the factors. In this case, we have three factors: $2$ , $6$ , and $3$ .

Step 3: Apply the exponent 9 to each factor:

$2^9$
$6^9$
$3^9$

Step 4: Combine these into the fully expanded form:

$2^9 \cdot 6^9 \cdot 3^9$ .

Review the answer options and note that there is no choice that matches this correct form.

Therefore, none of the answer choices are correct.

Final Answer

None of the answers are correct.

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to each factor separately: $(abc)^n = a^n \cdot b^n \cdot c^n$
Technique: $(2 \cdot 6 \cdot 3)^9 = 2^9 \cdot 6^9 \cdot 3^9$
Check: Every factor must have the same exponent applied ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only some factors
Don't write $2 \times 6^9 \times 3^9$ or $2^9 \times 6 \times 3^9$ = incomplete expansion! This violates the power of a product rule and gives incorrect results. Always apply the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to every factor?

The power of a product rule states that when you raise a product to a power, you must raise each factor to that power. This is like distributing multiplication - the exponent affects everything inside the parentheses!

What if I forget to apply the exponent to one factor?

Your answer will be completely wrong! For example, $2 \times 6^9 \times 3^9$ gives a much smaller result than the correct $2^9 \times 6^9 \times 3^9$ because you're missing $2^8$ as a factor.

Can I simplify the expression inside first?

Yes! You could calculate $2 \cdot 6 \cdot 3 = 36$ first, then find $36^9$ . Both methods give the same answer, but expanding first shows the structure more clearly.

How do I know if all the answer choices are wrong?

Compare each choice to your correct expansion. If none match $2^9 \times 6^9 \times 3^9$ , then "None of the answers are correct" is the right choice!

Does the order of factors matter in my answer?

No! Multiplication is commutative, so $2^9 \times 6^9 \times 3^9$ , $6^9 \times 2^9 \times 3^9$ , and other arrangements are all equivalent.

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