8.7- Powers and Roots of Negative Numbers

Today we started off class by going over the quizzes that we took on Friday. The class did very well with the average around a 89%. The quiz covered 8.4-8.6. Some key concepts that were on the quiz for 8.4 included being able to on your calculator, convert between radical exponent notation and rational exponent notation, simplify nested radicals and compute geometric mean. For 8.5 the important concepts were to simplify using factor tree or the "jailbreak" method. Lastly, 8.6 the important concepts were to be able to simplify fractions with radicals in the denominator. So after the quiz Mr. Cope passed out a note sheet as usual and we got started with 8.7 the new lesson.

The first thing that we covered was Integer Powers of Negative Numbers:
- For the even powers the answer was a positive number
Ex:

= 81

=9

-For the odd powers the answer was a negative number
Ex:

= -27
= -3

- And of course anything raised to the 0 is always 1:

= 1

We also worked with negative powers and saw how the answer varied if they were included:
- We found out that the same result happens with if the power is odd then you get a negative answer and if the power is even then it is a positive solution, but there is one twist you evaluate the problem from the denominator spot as you recall from chapter 7.
Ex:
Even Powers:

=

=

Odd Powers:

=

=

The next concept of the lesson was the question was posed: Are there Noninteger powers of negative numbers?

Lets try:
Positive- = - alright positive radicals are ok. now lets try negative

Negative- - lets break it down = * * which is equal to i
*Remember- imaginary numbers = i

Now lets try the same sort of math multiplying * when x and y are positive and negative:

Ex:
Positive-

*=
Negative-

* = 3i * 2i = 6which is -6
*Remember = -1 and so for example is basicallyand = i so it is 3i

Next we went over the fact that you are not allowed to do negative numbers raised to a fraction
ex-

- this problem is impossible because you end up with 2 different solutions, instead we just call this undefined.

*But we did figure out a easy was around this was to use- form
ex-

= -3

Then we asked ourselves what happened in form if x is negative and n is odd-
The answer is that you get one solution just how the problem above worked out-
= -3


The explanation to this is in the graph where there is a single horizontal line going through the y axis: this signifies 1 solution-

Last we saw what happens when x is negative and n is even
We found that it is undefined you can see this on the graph with a parabola and there is a single horizontal line going through the y axis: this signifies 0 solutions or undefined-





If this still doesn't explain the concept well enough the summary on the back of the 8.7 notes does a great job on stating the conclusion to the certain problems are.