Compare Absolute Values: Analyzing -|-81| and |9²|

Q: What's the difference between |-81| and -|-81|?

Big difference! $ |-81| = 81 $ (positive), but $ -|-81| = -81 $ (negative). The negative sign outside the absolute value bars changes the final result.

Q: Why do I calculate 9² before taking the absolute value?

Follow order of operations! Inside absolute value bars, exponents come first: $ 9^2 = 81 $, then $ |81| = 81 $. Always work from inside out.

Q: How do I know if two numbers are opposites?

Two numbers are opposites if they have the same absolute value but different signs. Like -81 and 81: both have absolute value 81, but one is positive and one is negative.

Q: Can absolute value ever be negative?

Never! Absolute value is always positive or zero. But expressions like $ -|x| $ can be negative because the negative sign is outside the absolute value.

Q: What if I get confused about the order?

Remember: absolute value bars work like parentheses. Do everything inside first, then apply any operations outside. Write each step to avoid mistakes!

Absolute Value Operations with Negative Signs

Are the numbers opposite?

$-\left|-81\right|,\left|9^2\right|$

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Understand the problem

Are the numbers opposite?

$-\left|-81\right|,\left|9^2\right|$

Step-by-step solution

To solve this problem, let's proceed with the required steps:

Step 1: Calculate the absolute value $\left|-81\right|$ .
$\left|-81\right| = 81$ .
Step 2: Apply a negative sign to $\left|-81\right|$ :
$-\left|-81\right| = -81$ .
Step 3: Calculate the absolute value $\left|9^2\right|$ .
First, evaluate $9^2 = 81$ , so $\left|81\right| = 81$ .
Step 4: Compare $-81$ and $81$ .
The numbers $-81$ and $81$ are indeed opposites since one is the negative of the other.

Therefore, the solution to the problem is Yes, the numbers are opposites.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Rule: Evaluate absolute value first, then apply external operations
Technique: $-|-81| = -81$ and $|9^2| = 81$
Check: Opposite numbers have same absolute value but different signs ✓

Common Mistakes

Avoid these frequent errors

Applying negative sign before absolute value
Don't calculate -|-81| as |81| = 81! The negative sign outside the absolute value bars applies after finding |−81| = 81. Always evaluate absolute value first, then apply the negative: -|-81| = -81.

Practice Quiz

Test your knowledge with interactive questions

Determine the absolute value of the following number:

$\left|18\right|=$

FAQ

Everything you need to know about this question

What's the difference between |-81| and -|-81|?

Big difference! $|-81| = 81$ (positive), but $-|-81| = -81$ (negative). The negative sign outside the absolute value bars changes the final result.

Why do I calculate 9² before taking the absolute value?

Follow order of operations! Inside absolute value bars, exponents come first: $9^2 = 81$ , then $|81| = 81$ . Always work from inside out.

How do I know if two numbers are opposites?

Two numbers are opposites if they have the same absolute value but different signs. Like -81 and 81: both have absolute value 81, but one is positive and one is negative.

Can absolute value ever be negative?

Never! Absolute value is always positive or zero. But expressions like $-|x|$ can be negative because the negative sign is outside the absolute value.

What if I get confused about the order?

Remember: absolute value bars work like parentheses. Do everything inside first, then apply any operations outside. Write each step to avoid mistakes!

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