Solve (4×2)² : Order of Operations with Squares

Exponent Rules with Product Expressions

Choose the expression that corresponds to the following:

(4×2)2= \left(4\times2\right)^2=

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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 containing a multiplication operation with an outer exponent
00:07 Raise each factor to the power
00:10 We will apply this formula to our exercise
00:14 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Choose the expression that corresponds to the following:

(4×2)2= \left(4\times2\right)^2=

2

Step-by-step solution

To solve the problem (4×2)2 \left(4\times2\right)^2, we need to apply the power of a product exponent rule. This rule states that (ab)n=an×bn (ab)^n = a^n \times b^n .

Here, a=4 a = 4 , b=2 b = 2 and n=2 n = 2 .

  • Step 1: Apply the power of a product rule: (4×2)2=42×22 \left(4 \times 2\right)^2 = 4^2 \times 2^2 .

Thus, the expression (4×2)2 \left(4\times2\right)^2 is equivalent to 42×22 4^2 \times 2^2 .

3

Final Answer

42×22 4^2\times2^2

Key Points to Remember

Essential concepts to master this topic
  • Power of Product Rule: (ab)n=an×bn (ab)^n = a^n \times b^n for any numbers
  • Technique: (4×2)2=42×22=16×4=64 (4\times2)^2 = 4^2 \times 2^2 = 16 \times 4 = 64
  • Check: Calculate inside parentheses first: (8)2=64 (8)^2 = 64 matches our answer ✓

Common Mistakes

Avoid these frequent errors
  • Calculating inside parentheses before applying the exponent rule
    Don't solve (4×2)2 (4\times2)^2 as 82 8^2 first = you miss the power of product pattern! This gives the right numerical answer but wrong algebraic form. Always recognize when to apply (ab)n=an×bn (ab)^n = a^n \times b^n to match the required expression format.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't I just multiply 4×2=8 first and then square it?

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You can do that for the numerical answer, but this question asks for the expression form. The power of product rule (ab)n=an×bn (ab)^n = a^n \times b^n shows the algebraic structure needed.

How do I remember the power of product rule?

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Think: "The exponent applies to each factor inside the parentheses." So (4×2)2 (4\times2)^2 means both 4 and 2 get squared: 42×22 4^2 \times 2^2 .

What's the difference between 42×22 4^2\times2^2 and 82×22 8^2\times2^2 ?

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42×22=16×4=64 4^2\times2^2 = 16\times4 = 64 while 82×22=64×4=256 8^2\times2^2 = 64\times4 = 256 . The first keeps the original factors separate, which is what the power of product rule requires.

Does this rule work with more than two numbers?

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Yes! For example, (2×3×5)2=22×32×52 (2\times3\times5)^2 = 2^2\times3^2\times5^2 . The exponent applies to every single factor inside the parentheses.

Can I use this rule backwards?

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Absolutely! If you see 34×74 3^4\times7^4 , you can write it as (3×7)4 (3\times7)^4 . This is called factoring out the common exponent.

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