Solve -(-5)³: Evaluating the Negative of a Negative Cube

Q: Why does $ (-5)^3 $ equal -125 instead of +125?

Because the exponent 3 is odd! When you multiply an odd number of negative factors, the result stays negative: $ (-5) \times (-5) \times (-5) = 25 \times (-5) = -125 $.

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

$ -5^3 = -(5^3) = -125 $ (negative applied to 5 cubed), while $ (-5)^3 = (-5) \times (-5) \times (-5) = -125 $ (negative 5 cubed). They equal the same thing, but the parentheses show the negative is part of the base.

Q: How do I remember when two negatives make a positive?

Think of it as opposite directions! A negative of a negative brings you back to positive, just like turning around twice brings you back to your original direction.

Q: Why don't we get $ -125 $ as the final answer?

Because there's an outer negative sign in front of the parentheses! After getting $ (-5)^3 = -125 $, we still need to apply that outer negative: $ -(-125) = +125 $.

Q: What would happen if the exponent was even instead?

If we had $ -(-5)^2 $, then $ (-5)^2 = +25 $ (even exponent makes it positive), so $ -(-5)^2 = -(+25) = -25 $. The final answer would be negative!

Order of Operations with Nested Negative Signs

$-(-5)^3=$

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Step-by-step video solution

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00:00 Solve

00:04 First, let's calculate the sign

00:09 Odd power, therefore the sign will be negative

00:21 Let's calculate the power

00:33 Negative times negative is always positive

00:37 And this is the solution to the question

Step-by-step written solution

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Understand the problem

$-(-5)^3=$

Step-by-step solution

To solve $-(-5)^3$ , we proceed as follows:

Step 1: Calculate $(-5)^3$ . Since the exponent is 3, which is odd, the result will be negative. Specifically:
$(-5)^3 = (-5) \times (-5) \times (-5)$ .
Step 2: Compute the multiplication:
$(-5) \times (-5) = 25$ (as multiplying two negatives gives a positive).
$25 \times (-5) = -125$ (as multiplying positive by negative gives a negative).
Step 3: Now apply the outer negative to this result:
$-(-125) = 125$ .

Therefore, the solution to the problem is $125$ .

Final Answer

$125$

Key Points to Remember

Essential concepts to master this topic

Order of Operations: Calculate exponents first, then apply outer negative signs
Technique: $(-5)^3 = -125$ , then apply negative: $-(-125) = 125$
Check: Verify odd exponents give negatives, then two negatives make positive ✓

Common Mistakes

Avoid these frequent errors

Applying the outer negative sign before calculating the exponent
Don't calculate $-(-5)^3$ as $(-(-5))^3 = 5^3 = 125$ ! This changes the base from -5 to +5, giving the wrong calculation path. Always calculate the exponent $(-5)^3$ first, then apply the outer negative sign.

Practice Quiz

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

FAQ

Everything you need to know about this question

Why does $(-5)^3$ equal -125 instead of +125?

Because the exponent 3 is odd! When you multiply an odd number of negative factors, the result stays negative: $(-5) \times (-5) \times (-5) = 25 \times (-5) = -125$ .

What's the difference between $-5^3$ and $(-5)^3$ ?

$-5^3 = -(5^3) = -125$ (negative applied to 5 cubed), while $(-5)^3 = (-5) \times (-5) \times (-5) = -125$ (negative 5 cubed). They equal the same thing, but the parentheses show the negative is part of the base.

How do I remember when two negatives make a positive?

Think of it as opposite directions! A negative of a negative brings you back to positive, just like turning around twice brings you back to your original direction.

Why don't we get $-125$ as the final answer?

Because there's an outer negative sign in front of the parentheses! After getting $(-5)^3 = -125$ , we still need to apply that outer negative: $-(-125) = +125$ .

What would happen if the exponent was even instead?

If we had $-(-5)^2$ , then $(-5)^2 = +25$ (even exponent makes it positive), so $-(-5)^2 = -(+25) = -25$ . The final answer would be negative!

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Solve -(-5)³: Evaluating the Negative of a Negative Cube

Order of Operations with Nested Negative Signs

❤️ 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 $(-5)^3$ equal -125 instead of +125?

What's the difference between $-5^3$ and $(-5)^3$ ?

How do I remember when two negatives make a positive?

Why don't we get $-125$ as the final answer?

What would happen if the exponent was even instead?

🌟 Unlock Your Math Potential

More Questions

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Evaluate -6²: Understanding Negative Numbers and Exponents

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Solve -(-5)³: Evaluating the Negative of a Negative Cube

Order of Operations with Nested Negative Signs

❤️ 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 (−5)3 (-5)^3 (−5)3 equal -125 instead of +125?

What's the difference between −53 -5^3 −53 and (−5)3 (-5)^3 (−5)3?

How do I remember when two negatives make a positive?

Why don't we get −125 -125 −125 as the final answer?

What would happen if the exponent was even instead?

🌟 Unlock Your Math Potential

More Questions

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Evaluate -6²: Understanding Negative Numbers and Exponents

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Why does $(-5)^3$ equal -125 instead of +125?

What's the difference between $-5^3$ and $(-5)^3$ ?

Why don't we get $-125$ as the final answer?