The document discusses powers of negative numbers. Some key points are: 1) The even powers of -2 are equal to the same power of 2, while the odd powers of -2 are the negative of the same power of 2. 2) The powers of -1 follow a pattern of (-1)1 = -1, (-1)2 = 1, (-1)3 = -1, and so on. 3) To write fractions like 42/45, we define negative powers using a new meaning - that x(-n) = 1/xn, to allow expressions like 4(-3) to have meaning as 1/43.

1. Mathematics
Submitted By:
MEENU M
KUCTE, Kumarapuram

2. Standard: viii
Unit: Negative Numbers

3. Powers of negatives

4. What are the powers of 2?
21 = 2
22 = 2*2 = 4
23 = 2*2*2 = 4*2 = 8
………………………………………
…………………………………………………

5. What about the powers of (-2)?
(-2)1 = (-2)
(-2)2 = (-2)*(-2) = 4
(-2)3 = (-2)*(-2)*(-2) = 4*(-2) =(-8)
………………………………………
…………………………………………………

6. Thus every even power of (-2)
is equal to the same power of 2;
every odd power of (-2) is the
negative of the same power of 2.

7. It is true for other number also…
Isn’t it ?
So what are the power of (-1)?
 (-1)1 = (-1)
 (-1)2 = 1
 (-1)3 = (-1)
………………………………………

8. Now we can consider fraction…
How can we write 45/42
45/42 = 45-2
= 43

9. How can we write 42/45
42/45 = 4(2-5)
= 4(-3)
According to our definition of
powers, does 4(-3) have any
meaning?

10. What is the meaning in saying ,
the product of (-3) fours?
So we give a new meaning to
negative powers, different from
repeated multiplication.

11. If we want to get
42/45 = 4(2-5)
= 4(-3),
then we should define
4(-3) = 1/43

12. In general we make the following definition,
for all x≠0 and for all natural
number n
x(-n) = 1/xn

13. We can combine two of the general
principles on the quotients
of powers to a single principles namely,
xm/xn = xm-n
whether m>n or m<n

14. Thank you