Multiply Powers: Solving (-1)^99 × (-1)^9

Q: Why does (-1) raised to any even power equal 1?

Think of it this way: $ (-1)^2 = (-1) \times (-1) = 1 $. Each pair of negative ones multiplies to give positive 1. Since even numbers can always be paired up, you always get 1!

Q: Can I use the rule a^m × a^n = a^(m+n) with negative bases?

Absolutely! This exponent rule works for all bases, including negative ones. So $ (-1)^{99} \cdot (-1)^9 = (-1)^{99+9} = (-1)^{108} $.

Q: How do I remember when (-1) to a power is positive or negative?

Easy trick: Odd powers are negative, even powers are positive. Think "Odd = negative One" and "Even = positive onE"!

Q: What if the exponents were different, like (-1)^50 × (-1)^7?

Same process! Add exponents: 50 + 7 = 57. Since 57 is odd, the answer would be -1. The key is always checking if your final exponent is odd or even.

Q: Do I always need to add the exponents when multiplying powers?

Yes, when the bases are the same! The rule $ a^m \times a^n = a^{m+n} $ is one of the most important exponent rules you'll use.

Exponent Rules with Negative Base Powers

$(-1)^{99}\cdot(-1)^9=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 First let's calculate the signs

00:07 Odd power, therefore the sign remains negative

00:21 This applies to both powers

00:29 1 to any power is always equal to 1

00:43 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

$(-1)^{99}\cdot(-1)^9=$

Step-by-step solution

To solve this problem, we need to evaluate the expression $(-1)^{99} \cdot (-1)^9$ .

The first step is to evaluate each component:

Step 1: Evaluate $(-1)^{99}$ .
Since 99 is an odd number, $(-1)^{99} = -1$ . This is because any odd power of $-1$ results in $-1$ .
Step 2: Evaluate $(-1)^9$ .
Similarly, since 9 is also an odd number, $(-1)^9 = -1$ . Again, an odd exponent means the result is $-1$ .

Step 3: Multiply the results from step 1 and step 2:
$(-1)^{99} \cdot (-1)^9 = (-1) \cdot (-1) = 1$ .

Thus, the value of the expression is $\boxed{1}$ .

Final Answer

$1$

Key Points to Remember

Essential concepts to master this topic

Rule: Odd powers of (-1) equal -1, even powers equal 1
Technique: Use exponent rule: $(-1)^{99} \cdot (-1)^9 = (-1)^{99+9} = (-1)^{108}$
Check: Since 108 is even, $(-1)^{108} = 1$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check if the total exponent is odd or even
Don't just look at individual exponents like 99 and 9 = thinking the answer is negative! This ignores the multiplication rule. Always add exponents first: 99 + 9 = 108, then check if 108 is even or odd.

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

Why does (-1) raised to any even power equal 1?

Think of it this way: $(-1)^2 = (-1) \times (-1) = 1$ . Each pair of negative ones multiplies to give positive 1. Since even numbers can always be paired up, you always get 1!

Can I use the rule a^m × a^n = a^(m+n) with negative bases?

Absolutely! This exponent rule works for all bases, including negative ones. So $(-1)^{99} \cdot (-1)^9 = (-1)^{99+9} = (-1)^{108}$ .

How do I remember when (-1) to a power is positive or negative?

Easy trick: Odd powers are negative, even powers are positive. Think "Odd = negative One" and "Even = positive onE"!

What if the exponents were different, like (-1)^50 × (-1)^7?

Same process! Add exponents: 50 + 7 = 57. Since 57 is odd, the answer would be -1. The key is always checking if your final exponent is odd or even.

Do I always need to add the exponents when multiplying powers?

Yes, when the bases are the same! The rule $a^m \times a^n = a^{m+n}$ is one of the most important exponent rules you'll use.

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