Complete the Expression: (3×6×4)^(2a) Power Operation

Power Distribution with Multiple Factors

Insert the corresponding expression:

(3×6×4)2a= \left(3\times6\times4\right)^{2a}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Simplify the following problem
00:04 In order to open parentheses with a multiplication operation and an outside exponent
00:08 Raise each factor to the power
00:15 We will apply this formula to our exercise
00:19 Note that all the factors in the multiplication operation have the same exponent (N)
00:24 Therefore we will raise each factor to this power
00:30 This is the solution

Step-by-step written solution

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1

Understand the problem

Insert the corresponding expression:

(3×6×4)2a= \left(3\times6\times4\right)^{2a}=

2

Step-by-step solution

To simplify the expression (3×6×4)2a(3 \times 6 \times 4)^{2a}, we apply the power of a product rule:

(3×6×4)2a=32a×62a×42a (3 \times 6 \times 4)^{2a} = 3^{2a} \times 6^{2a} \times 4^{2a}

This expression tells us that the exponent 2a2a is distributed to each factor inside the parentheses.

Therefore, the correct answer is 32a×62a×42a3^{2a} \times 6^{2a} \times 4^{2a}, corresponding to choice C.

3

Final Answer

32a×62a×42a 3^{2a}\times6^{2a}\times4^{2a}

Key Points to Remember

Essential concepts to master this topic
  • Power of Product Rule: Distribute exponent to each factor inside parentheses
  • Technique: (a×b×c)n=an×bn×cn (a \times b \times c)^n = a^n \times b^n \times c^n
  • Check: Each factor gets the same exponent 2a 2a applied ✓

Common Mistakes

Avoid these frequent errors
  • Applying exponent to only some factors
    Don't apply 2a 2a to just one or two factors like 32a×6×4 3^{2a} \times 6 \times 4 = incomplete distribution! This violates the power of product rule and gives wrong results. Always distribute the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why does the exponent go to all three numbers?

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The power of product rule says when you raise a product to a power, every factor gets raised to that power. Think of it like this: (3×6×4)2a (3 \times 6 \times 4)^{2a} means multiplying (3×6×4) (3 \times 6 \times 4) by itself 2a 2a times!

What if I calculated 3×6×4=72 3 \times 6 \times 4 = 72 first?

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You could write 722a 72^{2a} , but that's not what the question asks for. The question wants you to show the distributed form using the power rule: 32a×62a×42a 3^{2a} \times 6^{2a} \times 4^{2a} .

Does it matter what order I write the factors?

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No! Since multiplication is commutative, you can write the factors in any order. 32a×62a×42a 3^{2a} \times 6^{2a} \times 4^{2a} equals 62a×32a×42a 6^{2a} \times 3^{2a} \times 4^{2a} .

What if the exponent was just a number like 3?

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The same rule applies! (3×6×4)3=33×63×43 (3 \times 6 \times 4)^3 = 3^3 \times 6^3 \times 4^3 . Whether the exponent is a number, variable, or expression like 2a 2a , it gets distributed to every factor.

How do I remember this rule?

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Think 'distribute the power' - just like distributing treats to friends, the exponent gets distributed to every factor inside the parentheses. No factor gets left out!

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