Calculate -3³ vs (-3)³: Understanding Negative Number Exponents

Q: What's the difference between -3³ and (-3)³?

Great question! -3³ means "the opposite of 3 cubed" = -(3³) = -27. But (-3)³ means "negative 3, all cubed" = (-3) × (-3) × (-3) = -27. They happen to equal the same here, but the process is different!

Q: Why do both expressions equal -27?

This is special for odd exponents! For -3³: we get -(27) = -27. For (-3)³: negative times negative times negative = -27. With even exponents like 2, you'd get different results!

Q: How do I remember when to use parentheses?

Parentheses show what gets raised to the power. If you want the negative number cubed, use (-3)³. If you want 3 cubed, then made negative, write -3³.

Q: What if the exponent was even, like 2?

Then you'd see a difference! -3² = -9 (negative of 9), but (-3)² = 9 (positive, since negative times negative is positive).

Q: Do I add or subtract -27 and -27?

You add them! The expression is -27 + (-27). When adding negative numbers, you go further in the negative direction: -27 + (-27) = -54.

Q: How can I check my work?

Calculate each part separately first: $ -3^3 = -27 $ and $ (-3)^3 = -27 $. Then add: $ -27 + (-27) = -54 $. Double-check by working step by step!

Exponent Operations with Parentheses Placement

$-3^3+(-3)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Let's calculate the power without the sign

00:09 Now let's calculate the sign of the second power

00:12 Odd power, therefore the sign remains negative

00:26 Let's calculate the power and continue solving

00:37 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

$-3^3+(-3)^3=$

Step-by-step solution

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

Step 1: Calculate $-3^3$ and $(-3)^3$ .
Step 2: Add the results from both calculations.

Now, let's break this down further:
Step 1: First, compute $-3^3$ .
The expression $-3^3$ should be interpreted as $- (3^3)$ . This means we first calculate $3^3$ , which is $3 \times 3 \times 3 = 27$ . The negative sign in front gives us $-27$ .
Next, calculate $(-3)^3$ .
Here, the base is $-3$ , so we calculate $(-3) \times (-3) \times (-3)$ . This gives us:
- Multiply the first two factors: $(-3) \times (-3) = 9$ .
- Multiply the result by the last factor: $9 \times (-3) = -27$ .
So, $(-3)^3 = -27$ .

Step 2: Add the results $-27 + (-27)$ .
This computation is $-27 - 27 = -54$ .

Therefore, the solution to the problem is $-54$ .

Final Answer

$-54$

Key Points to Remember

Essential concepts to master this topic

Order of Operations: Exponents come before multiplication and subtraction
Technique: $-3^3 = -(3^3) = -27$ while $(-3)^3 = -27$
Check: Verify each calculation separately, then add: -27 + (-27) = -54 ✓

Common Mistakes

Avoid these frequent errors

Treating -3³ and (-3)³ as the same expression
Don't assume -3³ equals (-3)³ = they're different! Without parentheses, the negative sign isn't part of the base being cubed. Always remember -3³ means -(3³) = -27, while (-3)³ means the negative number cubed.

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

What's the difference between -3³ and (-3)³?

Great question! -3³ means "the opposite of 3 cubed" = -(3³) = -27. But (-3)³ means "negative 3, all cubed" = (-3) × (-3) × (-3) = -27. They happen to equal the same here, but the process is different!

Why do both expressions equal -27?

This is special for odd exponents! For -3³: we get -(27) = -27. For (-3)³: negative times negative times negative = -27. With even exponents like 2, you'd get different results!

How do I remember when to use parentheses?

Parentheses show what gets raised to the power. If you want the negative number cubed, use (-3)³. If you want 3 cubed, then made negative, write -3³.

What if the exponent was even, like 2?

Then you'd see a difference! -3² = -9 (negative of 9), but (-3)² = 9 (positive, since negative times negative is positive).

Do I add or subtract -27 and -27?

You add them! The expression is -27 + (-27). When adding negative numbers, you go further in the negative direction: -27 + (-27) = -54.

How can I check my work?

Calculate each part separately first: $-3^3 = -27$ and $(-3)^3 = -27$ . Then add: $-27 + (-27) = -54$ . Double-check by working step by step!

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