Compare Expressions: -2² ⬜ (-2)³ | Finding the Missing Operator

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

In -(2)², you square 2 first to get 4, then apply the negative to get -4. In (-2)², you're squaring the entire negative number, so (-2) × (-2) = +4. The parentheses make all the difference!

Q: Why is -4 greater than -8?

On the number line, -4 is to the right of -8, making it larger. Think of it like temperature: -4°F is warmer (greater) than -8°F. The closer a negative number is to zero, the greater it is.

Q: How do I remember the order of operations with negative signs?

Use PEMDAS! Parentheses and Exponents come before applying that outside negative sign. Think: "Do the power first, then flip the sign."

Q: What if the exponent is even vs odd with negative bases?

With negative bases: even exponents give positive results $ (-2)^2 = 4 $, odd exponents give negative results $ (-2)^3 = -8 $. This is because you multiply an even or odd number of negative signs!

Q: How can I double-check my comparison?

Plot both numbers on a number line or think about real situations. Which is warmer: -4° or -8°? Which is a smaller debt: owing $4 or owing $8? The answer helps confirm -4 > -8.

Negative Number Exponents with Order of Operations

Which is larger?

$-(2)^2⬜(-2)^3$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Place the appropriate sign

00:03 Let's calculate the power without the sign, the sign is not part of the power

00:07 Now let's calculate the sign of the second power

00:10 Odd power, therefore the sign remains negative

00:16 Let's calculate the power and compare

00:21 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?

$-(2)^2⬜(-2)^3$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Calculate $-(2)^2$ .
Here, $(2)^2 = 4$ , so $-(2)^2 = -4$ .
Step 2: Calculate $(-2)^3$ .
Since $(-2)^3 = (-2) \times (-2) \times (-2) = -8$ .
Step 3: Compare the results.
We have $-(2)^2 = -4$ and $(-2)^3 = -8$ . Comparing -4 and -8, we see that $-4 > -8$ .

Since $-4$ is greater than $-8$ , the symbol $>$ is correct.

Therefore, the solution to the problem is $>$ .

Final Answer

$>$

Key Points to Remember

Essential concepts to master this topic

Order of Operations: Apply exponent first, then handle negative sign
Technique: $-(2)^2 = -(4) = -4$ vs $(-2)^3 = -8$
Check: Compare final values: -4 is greater than -8 because -4 > -8 ✓

Common Mistakes

Avoid these frequent errors

Applying the negative sign before the exponent
Don't calculate -(2)² as (-2)² = 4! The negative sign stays outside until after exponentiation is complete. This changes -4 to +4, making your comparison completely wrong. Always follow order of operations: exponent first, then apply the negative sign.

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

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

In -(2)², you square 2 first to get 4, then apply the negative to get -4. In (-2)², you're squaring the entire negative number, so (-2) × (-2) = +4. The parentheses make all the difference!

Why is -4 greater than -8?

On the number line, -4 is to the right of -8, making it larger. Think of it like temperature: -4°F is warmer (greater) than -8°F. The closer a negative number is to zero, the greater it is.

How do I remember the order of operations with negative signs?

Use PEMDAS! Parentheses and Exponents come before applying that outside negative sign. Think: "Do the power first, then flip the sign."

What if the exponent is even vs odd with negative bases?

With negative bases: even exponents give positive results $(-2)^2 = 4$ , odd exponents give negative results $(-2)^3 = -8$ . This is because you multiply an even or odd number of negative signs!

How can I double-check my comparison?

Plot both numbers on a number line or think about real situations. Which is warmer: -4° or -8°? Which is a smaller debt: owing $4 or owing$ 8? The answer helps confirm -4 > -8.

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