Comparing Absolute Values: Is -|-64| Equal to |8²|?

Absolute Value Operations with Negative Signs

Are these two numbers opposites?

−∣−64∣,∣82∣ -\left|-64\right|, \left|8^2\right|

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Are these two numbers opposites?

−∣−64∣,∣82∣ -\left|-64\right|, \left|8^2\right|

2

Step-by-step solution

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

  • Step 1: Calculate the absolute value of −64-64.
  • Step 2: Apply the negative sign to the absolute value calculated.
  • Step 3: Calculate the absolute value of 828^2.
  • Step 4: Compare the two numbers to determine if they are opposites.

Now, let's work through each step:

Step 1: Calculate ∣−64∣\left|-64\right|.
The absolute value of −64-64 is ∣−64∣=64|-64| = 64.

Step 2: Apply the negative sign to this absolute value.
−∣−64∣=−64 -\left|-64\right| = -64

Step 3: Calculate ∣82∣\left|8^2\right|.
First, compute 82=648^2 = 64. Then, calculate ∣64∣=64|64| = 64.

Step 4: Compare the numbers.
We have −64-64 from Step 2 and 6464 from Step 3. These two numbers are opposites because the opposite of 6464 is −64-64.

Therefore, the two numbers are indeed opposites.

3

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic
  • Order: Calculate absolute value first, then apply outside operations
  • Technique: −∣−64∣=−(64)=−64 -|-64| = -(64) = -64 and ∣82∣=∣64∣=64 |8^2| = |64| = 64
  • Check: Opposites have same absolute value but different signs: -64 and 64 ✓

Common Mistakes

Avoid these frequent errors
  • Applying the negative sign before calculating absolute value
    Don't calculate −∣−64∣ -|-64| as ∣64∣=64 |64| = 64 ! This ignores the negative sign outside the absolute value bars and gives the wrong result. Always calculate the absolute value first, then apply any operations outside the bars.

Practice Quiz

Test your knowledge with interactive questions

\( \left|-19\frac{1}{4}\right|= \)

FAQ

Everything you need to know about this question

What's the difference between |-64| and -|-64|?

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∣−64∣=64 |-64| = 64 (just the absolute value), but −∣−64∣=−64 -|-64| = -64 (negative sign applied after finding absolute value). The negative outside the bars changes the final result!

How do I know if two numbers are opposites?

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Two numbers are opposites if they have the same absolute value but different signs. For example, 64 and -64 are opposites because ∣64∣=∣−64∣=64 |64| = |-64| = 64 .

Do I calculate the exponent before or after the absolute value?

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Calculate the exponent first! In ∣82∣ |8^2| , compute 82=64 8^2 = 64 first, then find ∣64∣=64 |64| = 64 . Always follow the order of operations.

Can absolute values ever be negative?

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No! Absolute values are always positive or zero. However, you can have a negative sign outside the absolute value bars, like −∣x∣ -|x| , which makes the entire expression negative.

What if I get confused about which operation to do first?

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Remember PEMDAS! Calculate what's inside the absolute value bars first (like exponents), then find the absolute value, then apply any operations outside the bars (like negative signs).

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