Compare (-1)^100 and -1^100: Exponent Rule Challenge

Q: Why does (-1)^100 equal 1 but -1^100 equal -1?

The parentheses make all the difference! In $ (-1)^{100} $, the entire negative one is the base. Since 100 is even, multiplying -1 by itself 100 times gives a positive result. In $ -1^{100} $, only the 1 is raised to the 100th power, then we apply the negative sign.

Q: How do I remember when negative numbers raised to even powers are positive?

Think of it like this: negative × negative = positive. When you multiply an even number of negative values together, you get pairs that each make a positive result. Even exponents of negative numbers (in parentheses) are always positive!

Q: What if the exponent was odd instead of 100?

Great question! If we had $ (-1)^{99} $, the result would be -1 because odd powers of negative numbers stay negative. But $ -1^{99} $ would still be -1 for the same reason as before.

Q: Does this rule work with other negative numbers too?

Absolutely! For example: $ (-3)^4 = 81 $ (positive) but $ -3^4 = -81 $ (negative). The parentheses rule applies to any negative base with any exponent.

Q: How can I avoid making this mistake on tests?

Look for parentheses around negative numbersWrite out the first few multiplications to see the patternRemember: parentheses = entire negative baseNo parentheses = negative sign applied after exponentiation

Order of Operations with Parentheses and Exponents

Which is larger?

$(-1)^{100}⬜-1^{100}$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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

00:07 Even power, therefore the sign will be positive

00:18 In this number the power doesn't affect the sign, therefore negative

00:26 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?

$(-1)^{100}⬜-1^{100}$

Step-by-step solution

To determine which is larger between $(-1)^{100}$ and $-1^{100}$ , follow these steps:

Step 1: Evaluate $(-1)^{100}$ .
Since 100 is an even number, $(-1)^{100}$ simplifies to $(-1)$ multiplied by itself 100 times. Even powers of -1 result in $1$ , so $(-1)^{100} = 1$ .
Step 2: Evaluate $-1^{100}$ .
Notice that $-1^{100}$ is simply putting a negative sign in front of $1^{100}$ . Since powers of 1 are always 1, $1^{100} = 1$ , resulting in $-1^{100} = -1$ .
Step 3: Compare the results.
From our calculations, $(-1)^{100} = 1$ and $-1^{100} = -1$ . Comparing these, $1 > -1$ .

Thus, the expression $(-1)^{100}$ is greater than $-1^{100}$ .

Therefore, the correct comparison is $(-)^{100} > -1^{100}$ .

The correct choice from the possible answers is $>$ .

Final Answer

$>$

Key Points to Remember

Essential concepts to master this topic

Rule: Parentheses change the base of exponentiation completely
Technique: $(-1)^{100} = 1$ but $-1^{100} = -1$
Check: Even exponents of negative numbers in parentheses give positive results ✓

Common Mistakes

Avoid these frequent errors

Ignoring parentheses in exponential expressions
Don't treat (-1)^100 the same as -1^100 = both equal -1! Without parentheses, only the 1 gets raised to the 100th power, then made negative. Always recognize that parentheses make the entire negative number the base.

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

Why does (-1)^100 equal 1 but -1^100 equal -1?

The parentheses make all the difference! In $(-1)^{100}$ , the entire negative one is the base. Since 100 is even, multiplying -1 by itself 100 times gives a positive result. In $-1^{100}$ , only the 1 is raised to the 100th power, then we apply the negative sign.

How do I remember when negative numbers raised to even powers are positive?

Think of it like this: negative × negative = positive. When you multiply an even number of negative values together, you get pairs that each make a positive result. Even exponents of negative numbers (in parentheses) are always positive!

What if the exponent was odd instead of 100?

Great question! If we had $(-1)^{99}$ , the result would be -1 because odd powers of negative numbers stay negative. But $-1^{99}$ would still be -1 for the same reason as before.

Does this rule work with other negative numbers too?

Absolutely! For example: $(-3)^4 = 81$ (positive) but $-3^4 = -81$ (negative). The parentheses rule applies to any negative base with any exponent.

How can I avoid making this mistake on tests?

Look for parentheses around negative numbers
Write out the first few multiplications to see the pattern
Remember: parentheses = entire negative base
No parentheses = negative sign applied after exponentiation

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Powers - special cases questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime