Evaluate (a×3)³: Expanding the Cube of a Product

Power Rules with Product Expressions

Insert the corresponding expression:

(a×3)3= \left(a\times3\right)^3=

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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 In order to open parentheses with a multiplication operation and an outside exponent
00:08 We'll raise each factor to the power
00:12 We'll apply this formula to our exercise
00:19 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

(a×3)3= \left(a\times3\right)^3=

2

Step-by-step solution

To solve the problem (a×3)3 (a \times 3)^3 , we'll apply the power of a product rule which states that (x⋅y)n=xn⋅yn(x \cdot y)^n = x^n \cdot y^n.

Step 1: Identify the individual factors within the parentheses. In this expression, aa and 3 are multiplied together and are being raised to the power of 3.

Step 2: Apply the power of a product property: Distribute the exponent of 3 to both aa and 3 inside the parentheses. We do so as follows:
(a×3)3=a3×33(a \times 3)^3 = a^3 \times 3^3

Step 3: Express the result clearly. The expression simplifies to:
a3×33a^3 \times 3^3.

Therefore, the correct answer to this problem is a3×33 a^3 \times 3^3 .

3

Final Answer

a3×33 a^3\times3^3

Key Points to Remember

Essential concepts to master this topic
  • Power of Product Rule: (xy)n=xn×yn(xy)^n = x^n \times y^n distributes exponents to each factor
  • Technique: Apply exponent 3 to both aa and 3: (a×3)3=a3×33(a \times 3)^3 = a^3 \times 3^3
  • Check: Each factor inside parentheses gets raised to the same power ✓

Common Mistakes

Avoid these frequent errors
  • Only applying exponent to one factor
    Don't raise just aa to the 3rd power = a3×3a^3 \times 3! This ignores the power rule and gives an incomplete answer. Always apply the exponent to every 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 apply to both parts?

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The power of a product rule says when you raise a multiplication to a power, each factor gets that power. Think of it as (a×3)3=(a×3)×(a×3)×(a×3)(a \times 3)^3 = (a \times 3) \times (a \times 3) \times (a \times 3), which gives you a3×33a^3 \times 3^3.

What's the difference between 333^3 and just 3?

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333^3 means 3 multiplied by itself 3 times, so 33=3×3×3=273^3 = 3 \times 3 \times 3 = 27. Just writing 3 means the exponent wasn't applied correctly!

Can I multiply out the answer further?

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Yes! You could write a3×33a^3 \times 3^3 as a3×27a^3 \times 27 or 27a327a^3, but the form a3×33a^3 \times 3^3 shows you applied the rule correctly.

Does this work with more factors?

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Absolutely! For example, (2xy)4=24×x4×y4(2xy)^4 = 2^4 \times x^4 \times y^4. The exponent applies to every single factor inside the parentheses.

What if there's a negative sign?

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Treat the negative as part of the expression! For example, (−2a)3=(−2)3×a3=−8a3(-2a)^3 = (-2)^3 \times a^3 = -8a^3. The negative sign gets cubed too.

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