what do you know about the sign of a negative integer raised to an odd power and even power
Look at a number such as -2. Raised to an even power (say 2) is -2*-2 = +4.
Or raised to 4 (another even power) is -2*-2*-2*-2 = +16. Both 4 and 16 are positive numbers.
Raise -2 to an odd power (say 3) and we have -2*-2*-2 = -8 (a negative answer).
Or raised to -5 is
-2*-2*-2*-2*-2 = -32 (a negative answer). You should be able to get the connection from these example.
When a negative integer is raised to an odd power, the result is always a negative number. On the other hand, when a negative integer is raised to an even power, the result is always a positive number.
To understand why this is the case, let's break it down step by step:
1. Negative integer raised to an odd power:
- Start with a negative integer, let's say -3.
- When you raise -3 to an odd power, such as -3^3 or -3^5, the negative sign remains intact.
- This happens because multiplying a negative number by itself an odd number of times results in a negative value.
- For example, -3^3 = -3 * -3 * -3 = -27.
2. Negative integer raised to an even power:
- Again, start with a negative integer, let's use -2 this time.
- When you raise -2 to an even power, like -2^2 or -2^4, the negative sign is eliminated.
- This occurs because multiplying a negative number by itself an even number of times cancels out the negative sign, resulting in a positive value.
- For example, -2^2 = -2 * -2 = 4.
In summary, when you raise a negative integer to an odd power, the result is negative, while raising it to an even power yields a positive result.