Solve (-2)²)²: Double Square of a Negative Number

Exponent Rules with Nested Powers

((−2)2)2= ((-2)^2)^2=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:03 We'll break down the exponent into multiplications, and solve
00:08 Negative times negative is always positive
00:22 We'll break down the exponent into multiplications, and solve
00:26 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

((−2)2)2= ((-2)^2)^2=

2

Step-by-step solution

To solve the given problem, we'll use rules of exponents and follow these steps:

  • Step 1: Calculate the inner power. Simplify (−2)2(-2)^2.
  • Step 2: Calculate the result by taking the square of the previous result.

Here's the detailed breakdown:


Step 1: Determine (−2)2(-2)^2.
Using the property of exponents, (−2)2=(−2)×(−2)=4(-2)^2 = (-2) \times (-2) = 4.

Step 2: Now, square the result from Step 1.
You are to calculate 424^2.
42=4×4=164^2 = 4 \times 4 = 16.

Therefore, the solution to the problem is 16\boxed{16}.

3

Final Answer

16 16

Key Points to Remember

Essential concepts to master this topic
  • Order: Always evaluate innermost parentheses and powers first
  • Technique: Calculate (−2)2=4 (-2)^2 = 4 , then 42=16 4^2 = 16
  • Check: Verify that negative times negative equals positive: (−2)×(−2)=4 (-2) \times (-2) = 4 ✓

Common Mistakes

Avoid these frequent errors
  • Applying the outer exponent to the negative sign
    Don't calculate ((−2)2)2 ((-2)^2)^2 as (−2)4=−16 (-2)^4 = -16 ! This ignores the parentheses and gives the wrong sign. Always work from the inside out, calculating (−2)2=4 (-2)^2 = 4 first, then 42=16 4^2 = 16 .

Practice Quiz

Test your knowledge with interactive questions

Solve the following expression:

\( \)\( (-8)^2= \)

FAQ

Everything you need to know about this question

Why is (−2)2 (-2)^2 positive but −22 -2^2 negative?

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Great question! Parentheses matter! (−2)2 (-2)^2 means "negative two, squared" = (−2)×(−2)=4 (-2) \times (-2) = 4 . But −22 -2^2 means "the opposite of two squared" = −(2×2)=−4 -(2 \times 2) = -4 .

Can I just multiply the exponents together?

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Not in this case! The power rule (am)n=amn (a^m)^n = a^{mn} works, but you still need to be careful with signs. Here, ((−2)2)2=(−2)2×2=(−2)4=16 ((-2)^2)^2 = (-2)^{2 \times 2} = (-2)^4 = 16 , which gives the same answer.

How do I know when a negative number raised to a power is positive?

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Even exponents make positives, odd exponents keep negatives! Since we have (−2)2 (-2)^2 (even power), the result is positive. Then 42 4^2 is also positive.

What's the difference between this and (−22)2 (-2^2)^2 ?

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Huge difference! (−22)2 (-2^2)^2 means (−(22))2=(−4)2=16 (-(2^2))^2 = (-4)^2 = 16 . But ((−2)2)2 ((-2)^2)^2 means (4)2=16 (4)^2 = 16 . They both equal 16, but for different reasons!

Why do I need to work from inside out?

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The order of operations requires it! Just like PEMDAS says to do parentheses first, you must evaluate the innermost expressions before moving outward. This prevents sign errors and ensures correct results.

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