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?

$(-1)^{100}⬜-1^{100}$

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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

00:07 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

Solve the following expression:

$$ $$ (-8)^2= $$

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

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