Simplify (2×8) Raised to Power (2y+2): Complete Expression

Exponent Rules with Product Bases

Insert the corresponding expression:

(2×8)2y+2= \left(2\times8\right)^{2y+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 with a multiplication operation and an outside exponent
00:09 Raise each factor to the power
00:12 We'll apply this formula to our exercise
00:16 Note that our exponent is comprised of an addition operation and a power (N)
00:19 Therefore we'll raise each factor to this power
00:26 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:

(2×8)2y+2= \left(2\times8\right)^{2y+2}=

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the expression as (2×8)2y+2(2 \times 8)^{2y+2}.
  • Step 2: Apply the "Power of a Product" rule: (a×b)n=an×bn(a \times b)^n = a^n \times b^n.
  • Step 3: Apply this to the bases 2 and 8 in the given expression.

Now, let's work through each step:
Step 1: In our problem, the expression (2×8)2y+2(2 \times 8)^{2y+2} needs to be expanded.
Step 2: According to the exponent rule, we can rewrite the expression as 22y+2×82y+22^{2y+2} \times 8^{2y+2}.
Step 3: We have applied the power to each individual base within the parentheses.

Therefore, the corresponding expression is 22y+2×82y+2 2^{2y+2} \times 8^{2y+2} .

3

Final Answer

22y+2×82y+2 2^{2y+2}\times8^{2y+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 distributes exponent to each factor
  • Apply Technique: (2×8)2y+2=22y+2×82y+2(2 \times 8)^{2y+2} = 2^{2y+2} \times 8^{2y+2}
  • Verify Method: Check that each base gets the same exponent 2y+22y+2 ✓

Common Mistakes

Avoid these frequent errors
  • Distributing exponent incorrectly to factors
    Don't write (2×8)2y+2=22y×82(2 \times 8)^{2y+2} = 2^{2y} \times 8^2 by splitting the exponent! This ignores the power of product rule and gives wrong results. Always apply the entire exponent 2y+22y+2 to each 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 can't I just multiply 2×8 first and then raise to the power?

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You absolutely can! (2×8)2y+2=162y+2(2 \times 8)^{2y+2} = 16^{2y+2} is correct, but the question asks for the expanded form using the power of product rule to show each base separately.

What's the difference between this and 22y×822^{2y} \times 8^2?

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That's splitting the exponent incorrectly! The power of product rule says every factor gets the entire exponent 2y+22y+2, not parts of it.

How do I remember the power of product rule?

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Think of it as "everyone gets the same power". When you have (ab)n(ab)^n, both a and b get raised to the power n.

Does this work with more than two factors?

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Yes! (abc)n=an×bn×cn(abc)^n = a^n \times b^n \times c^n. The rule works for any number of factors inside the parentheses.

What if the exponent was just a number instead of 2y+22y+2?

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The rule works exactly the same way! For example: (2×8)3=23×83=8×512=4096(2 \times 8)^3 = 2^3 \times 8^3 = 8 \times 512 = 4096.

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