Calculate (-1)^99: Evaluating Powers of Negative One

Q: Why does (-1)^99 equal -1 instead of 1?

Because 99 is an odd number! When you multiply (-1) by itself an odd number of times, you get a negative result. Think of it as: odd exponents = negative answer, even exponents = positive answer.

Q: Is there a pattern I can memorize?

Yes! For any power of -1: odd exponent gives -1, even exponent gives +1. So $ (-1)^{98} = 1 $ but $ (-1)^{99} = -1 $.

Q: What if the exponent is really big like 999?

The size doesn't matter - only whether it's odd or even! Since 999 is odd, $ (-1)^{999} = -1 $. The pattern never changes no matter how big the number gets.

Q: How do I quickly tell if a number is odd or even?

Look at the last digit! If it ends in 1, 3, 5, 7, or 9, it's odd. If it ends in 0, 2, 4, 6, or 8, it's even. So 99 ends in 9, making it odd.

Q: Can I just multiply (-1) × (-1) × (-1) all the way to 99?

You could, but that's way too much work! Using the odd/even pattern rule gives you the answer instantly. Smart math means finding shortcuts, not doing unnecessary calculations.

Powers of Negative One with Odd Exponents

$(-1)^{99}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 First let's calculate the sign

00:06 Odd power, therefore the sign will be negative

00:10 Now let's calculate the power, 1 raised to any power is always equal to 1

00:16 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

$(-1)^{99}=$

Step-by-step solution

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

Step 1: Determine the nature of the exponent (odd or even).
Step 2: Apply the specific rule for the power of $-1$ based on the exponent's nature.

Now, let's work through each step:
Step 1: The exponent is $99$ , which is odd.
Step 2: For the power of $-1$ , the rule states that if the exponent is odd, $(-1)^n = -1$ . Thus, $(-1)^{99} = -1$ .

Therefore, the solution to the problem is $-1$ .

Final Answer

$-1$

Key Points to Remember

Essential concepts to master this topic

Pattern Rule: Negative one raised to odd powers always equals negative one
Recognition: Check if exponent is odd like 99, then $(-1)^{99} = -1$
Verification: Count pattern: $(-1)^1 = -1$ , $(-1)^3 = -1$ , confirms odd gives negative ✓

Common Mistakes

Avoid these frequent errors

Confusing the sign pattern for odd and even exponents
Don't think $(-1)^{99} = 1$ because 99 is large = wrong positive result! Large numbers don't change the pattern. Always check if the exponent is odd (gives -1) or even (gives 1).

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

Why does (-1)^99 equal -1 instead of 1?

Because 99 is an odd number! When you multiply (-1) by itself an odd number of times, you get a negative result. Think of it as: odd exponents = negative answer, even exponents = positive answer.

Is there a pattern I can memorize?

Yes! For any power of -1: odd exponent gives -1, even exponent gives +1. So $(-1)^{98} = 1$ but $(-1)^{99} = -1$ .

What if the exponent is really big like 999?

The size doesn't matter - only whether it's odd or even! Since 999 is odd, $(-1)^{999} = -1$ . The pattern never changes no matter how big the number gets.

How do I quickly tell if a number is odd or even?

Look at the last digit! If it ends in 1, 3, 5, 7, or 9, it's odd. If it ends in 0, 2, 4, 6, or 8, it's even. So 99 ends in 9, making it odd.

Can I just multiply (-1) × (-1) × (-1) all the way to 99?

You could, but that's way too much work! Using the odd/even pattern rule gives you the answer instantly. Smart math means finding shortcuts, not doing unnecessary calculations.

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