Compare -(5³) and 5³: Finding the Missing Operator

Q: What's the difference between -(5³) and (-5)³?

Big difference! $ -(5^3) $ means take 5³ = 125, then make it negative = -125. But $ (-5)^3 $ means multiply (-5) × (-5) × (-5) = -125. They happen to equal the same here, but understand the difference!

Q: Why isn't -125 equal to 125?

Negative and positive numbers are never equal! Think of a number line: -125 is 125 units to the left of zero, while 125 is 125 units to the right. They're opposites!

Q: How do I remember which is bigger?

Remember: any positive number is always bigger than any negative number. On a number line, numbers get bigger as you move right. So 125 > 0 > -125.

Q: What if the exponent was even instead of odd?

Great question! If we had $ -(5^2) $ vs $ 5^2 $, we'd get -25 vs 25. The comparison method stays the same: calculate the exponent first, then apply signs.

Q: Can I use a calculator for this?

Yes, but be careful with parentheses! For $ -(5^3) $, enter it as -(5^3) or -1×(5^3). Make sure you don't accidentally calculate $ (-5)^3 $ instead.

Negative Exponents with Sign Comparisons

Which is larger?

$-(5^3)⬜5^3$

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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 First, let's calculate the sign

00:06 Odd exponent, therefore the sign remains negative

00:09 In the second number same number, but positive

00:12 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?

$-(5^3)⬜5^3$

Step-by-step solution

To solve this problem, we'll compare the expressions $-(5^3)$ and $5^3$ by calculating each separately and then determining which is larger.

Step 1: Calculate $5^3$ .
This is equal to $5 \times 5 \times 5 = 125$ .

Step 2: Calculate $-(5^3)$ .
Since $5^3 = 125$ , applying the negative sign gives us $-(5^3) = -125$ .

Step 3: Compare the values.
We have $-(5^3) = -125$ and $5^3 = 125$ .
Clearly, $-125 < 125$ .

Thus, the correct answer is that $-(5^3) \lt 5^3$ .

The correct choice for this problem is $<$ .

Final Answer

$<$

Key Points to Remember

Essential concepts to master this topic

Order of Operations: Calculate exponent first, then apply negative sign
Technique: $-(5^3) = -(125) = -125$
Check: Compare final values: -125 < 125, so < is correct ✓

Common Mistakes

Avoid these frequent errors

Applying negative sign before calculating exponent
Don't calculate $(-5)^3$ instead of $-(5^3)$ = gets -125 vs -125 but misses the concept! The negative is outside the exponent, not part of the base. Always calculate the 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 -(5³) and (-5)³?

Big difference! $-(5^3)$ means take 5³ = 125, then make it negative = -125. But $(-5)^3$ means multiply (-5) × (-5) × (-5) = -125. They happen to equal the same here, but understand the difference!

Why isn't -125 equal to 125?

Negative and positive numbers are never equal! Think of a number line: -125 is 125 units to the left of zero, while 125 is 125 units to the right. They're opposites!

How do I remember which is bigger?

Remember: any positive number is always bigger than any negative number. On a number line, numbers get bigger as you move right. So 125 > 0 > -125.

What if the exponent was even instead of odd?

Great question! If we had $-(5^2)$ vs $5^2$ , we'd get -25 vs 25. The comparison method stays the same: calculate the exponent first, then apply signs.

Can I use a calculator for this?

Yes, but be careful with parentheses! For $-(5^3)$ , enter it as -(5^3) or -1×(5^3). Make sure you don't accidentally calculate $(-5)^3$ instead.

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