Solve (-(-2)²)² - 2³: Multiple Exponents and Negative Numbers

Q: Why does $ (-2)^2 = 4 $ but $ -(-2)^2 = -4 $ ?

Great question! In $ (-2)^2 $, the parentheses include the negative, so both the negative sign and 2 get squared: $ (-2) \times (-2) = 4 $. But $ -(-2)^2 $ means the negative outside applies after squaring, giving us $ -(4) = -4 $.

Q: How do I keep track of all the parentheses and negatives?

Work from the inside out! Start with the innermost parentheses $ (-2)^2 $, then move outward step by step. Write down each step clearly: $ (-2)^2 = 4 $, then $ -4 $, then $ (-4)^2 = 16 $.

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

$ (-4)^2 $ means the entire negative number is squared: $ (-4) \times (-4) = 16 $. But $ -4^2 $ means square 4 first, then apply the negative: $ -(4^2) = -16 $. Parentheses make all the difference!

Q: Why do we subtract $ 2^3 $ at the end?

Because the original expression shows $ (-(-2)^2)^2 - 2^3 $! The minus sign tells us to subtract $ 2^3 = 8 $ from our result of 16, giving us $ 16 - 8 = 8 $.

Q: How can I check if my answer is right?

Substitute back into each step! Verify: $ (-2)^2 = 4 $ ✓, $ -(-2)^2 = -4 $ ✓, $ (-4)^2 = 16 $ ✓, $ 2^3 = 8 $ ✓, and finally $ 16 - 8 = 8 $ ✓. Each step should make sense!

Order of Operations with Nested Exponents

$(-(-2)^2)^2-2^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 First let's calculate the sign

00:06 Even power, therefore the sign will be positive

00:13 Let's calculate the powers

00:31 Even power, therefore the sign will be positive

00:38 Let's calculate the square, and continue calculating

00:42 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

$(-(-2)^2)^2-2^3=$

Step-by-step solution

To solve this problem, we need to carefully apply the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Let's evaluate the expression step-by-step:

Step 1: Begin with the expression inside the innermost parentheses: $(-2)^2$ . This means we first square $-2$ , which results in $4$ because $(-2) \times (-2) = 4$ .
Step 2: Now, consider the negative sign outside the squared term. So, evaluate $-(-2)^2$ , which simplifies to $-4$ .
Step 3: Next, we need to square $-4$ : $(-4)^2 = (-4) \times (-4) = 16$ .
Step 4: Now, calculate the second part of the expression, $2^3$ , which equals $8$ because $2 \times 2 \times 2 = 8$ .
Step 5: Finally, subtract the result from Step 4 from the result of Step 3: $16 - 8 = 8$ .

Thus, the solution to the problem is $8$ .

Final Answer

$8$

Key Points to Remember

Essential concepts to master this topic

Rule: Work from innermost parentheses outward using PEMDAS order
Technique: $(-2)^2 = 4$ , then $-(-2)^2 = -4$ , then $(-4)^2 = 16$
Check: Final calculation $16 - 8 = 8$ matches our answer ✓

Common Mistakes

Avoid these frequent errors

Applying negative signs incorrectly to exponents
Don't evaluate $-(-2)^2$ as $(-(-2))^2 = 2^2 = 4$ ! The negative outside affects the result of $(-2)^2$ , not the base itself. Always calculate the exponent first, then apply the outside negative: $-4 = -4$ .

Practice Quiz

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Solve the following expression:

$$ $$ (-8)^2= $$

FAQ

Everything you need to know about this question

Why does $(-2)^2 = 4$ but $-(-2)^2 = -4$ ?

Great question! In $(-2)^2$ , the parentheses include the negative, so both the negative sign and 2 get squared: $(-2) \times (-2) = 4$ . But $-(-2)^2$ means the negative outside applies after squaring, giving us $-(4) = -4$ .

How do I keep track of all the parentheses and negatives?

Work from the inside out! Start with the innermost parentheses $(-2)^2$ , then move outward step by step. Write down each step clearly: $(-2)^2 = 4$ , then $-4$ , then $(-4)^2 = 16$ .

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

$(-4)^2$ means the entire negative number is squared: $(-4) \times (-4) = 16$ . But $-4^2$ means square 4 first, then apply the negative: $-(4^2) = -16$ . Parentheses make all the difference!

Why do we subtract $2^3$ at the end?

Because the original expression shows $(-(-2)^2)^2 - 2^3$ ! The minus sign tells us to subtract $2^3 = 8$ from our result of 16, giving us $16 - 8 = 8$ .

How can I check if my answer is right?

Substitute back into each step! Verify: $(-2)^2 = 4$ ✓, $-(-2)^2 = -4$ ✓, $(-4)^2 = 16$ ✓, $2^3 = 8$ ✓, and finally $16 - 8 = 8$ ✓. Each step should make sense!

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Solve (-(-2)²)² - 2³: Multiple Exponents and Negative Numbers

Order of Operations with Nested Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does $(-2)^2 = 4$ but $-(-2)^2 = -4$ ?

How do I keep track of all the parentheses and negatives?

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

Why do we subtract $2^3$ at the end?

How can I check if my answer is right?

🌟 Unlock Your Math Potential

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Solve (-(-2)²)² - 2³: Multiple Exponents and Negative Numbers

Order of Operations with Nested Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does (−2)2=4 (-2)^2 = 4 (−2)2=4 but −(−2)2=−4 -(-2)^2 = -4 −(−2)2=−4?

How do I keep track of all the parentheses and negatives?

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

Why do we subtract 23 2^3 23 at the end?

How can I check if my answer is right?

🌟 Unlock Your Math Potential

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Why does $(-2)^2 = 4$ but $-(-2)^2 = -4$ ?

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

Why do we subtract $2^3$ at the end?