Compare (-4)^2)^2 vs -(4^2)^2: Exponent Order Challenge

Q: What's the difference between (-4)² and -4²?

Big difference! $ (-4)^2 = 16 $ because the negative sign is inside parentheses and gets squared. But $ -4^2 = -16 $ because only the 4 gets squared, then we apply the negative.

Q: Why do both expressions equal the same thing?

Both expressions end up squaring a negative number! $ ((-4)^2)^2 $ becomes $ 16^2 $, and $ (-(4)^2)^2 $ becomes $ (-16)^2 $. Since any negative number squared is positive, both equal 256.

Q: How do I keep track of all these parentheses?

Work from the inside out! Start with the innermost parentheses, calculate that value, then move to the next layer. Write each step clearly: $ (-4)^2 = 16 $, then $ (16)^2 = 256 $.

Q: Do I always need to show every step?

Yes, especially with nested exponents! Each step helps you avoid sign errors and parentheses mistakes. Plus, your teacher can see your thinking process even if you make a small calculation error.

Exponent Operations with Parentheses and Signs

Which is larger?

$((-4)^2)^2⬜(-(4)^2)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Place the appropriate sign

00:04 First, let's calculate the sign

00:07 Even power, so the sign will be positive, let's calculate the power

00:14 Now let's calculate the sign of the second power

00:18 Even power, so the sign will be positive, let's calculate the power

00:27 Any negative number squared equals a positive number

00:33 We can see that the numbers are equal

00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which is larger?

$((-4)^2)^2⬜(-(4)^2)^2$

Step-by-step solution

To solve this problem, we must calculate both expressions step-by-step:

First, consider the expression $(( -4)^2)^2$ :

Evaluate $(-4)^2$ :
$(-4)^2 = (-4) \times (-4) = 16$ .
Now, evaluate $16^2$ :
$16^2 = 16 \times 16 = 256$ .

The value of $(( -4)^2)^2$ is $256$ .

Next, consider the expression $(-(4)^2)^2$ :

Evaluate $(4)^2$ :
$4^2 = 4 \times 4 = 16$ .
Now, consider the negative sign: $-(4)^2 = -(16) = -16$ .
Evaluate $(-16)^2$ :
$(-16)^2 = (-16) \times (-16) = 256$ .

The value of $(-(4)^2)^2$ is $256$ .

Therefore, the two expressions are equal.

Conclusion: The correct choice is $(=)$ .

Final Answer

$=$

Key Points to Remember

Essential concepts to master this topic

Order: Work inside parentheses first, then apply outer exponents
Technique: $(-4)^2 = 16$ , then $16^2 = 256$
Check: Both expressions equal 256, so they are equal ✓

Common Mistakes

Avoid these frequent errors

Applying negative signs incorrectly with exponents
Don't ignore parentheses and calculate $-4^2 = -16$ instead of $(-4)^2 = 16$ ! Missing parentheses changes which part gets the exponent. Always work inside parentheses first, then apply outer operations.

Practice Quiz

Test your knowledge with interactive questions

$$ $$ -(2)^2= $$

FAQ

Everything you need to know about this question

What's the difference between (-4)² and -4²?

Big difference! $(-4)^2 = 16$ because the negative sign is inside parentheses and gets squared. But $-4^2 = -16$ because only the 4 gets squared, then we apply the negative.

Why do both expressions equal the same thing?

Both expressions end up squaring a negative number! $((-4)^2)^2$ becomes $16^2$ , and $(-(4)^2)^2$ becomes $(-16)^2$ . Since any negative number squared is positive, both equal 256.

How do I keep track of all these parentheses?

Work from the inside out! Start with the innermost parentheses, calculate that value, then move to the next layer. Write each step clearly: $(-4)^2 = 16$ , then $(16)^2 = 256$ .

What if the outer exponent was odd instead of even?

Great question! If the outer exponent were odd (like 3), the results would be different because an odd power preserves the sign. Even powers always make the result positive.

Do I always need to show every step?

Yes, especially with nested exponents! Each step helps you avoid sign errors and parentheses mistakes. Plus, your teacher can see your thinking process even if you make a small calculation error.

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