REAL NUMBERS EKSPONENT NUMBERS.

Presentasi berjudul: "REAL NUMBERS EKSPONENT NUMBERS."— Transcript presentasi:

1 REAL NUMBERS EKSPONENT NUMBERS

2 Properties of the Exponent Numbers
2  2  2  2  ...  2 Symbolized by 2n Factor n 3  3  3  3  ...  3 Symbolized by 3n Factor n 8  8  8  8  ...  8 Symbolized by 8n Factor n Defined by: 1) an = a a a a  a Factor n 2) a1 = a Hal.: 2 Isi dengan Judul Halaman Terkait

3 Multiplication of the Exponent Numbers
a  a  a  …  a      a a  a  …  a p factor number a q factor number a (p + q) factor number a means ap+q  ap  aq = ap+q Example : x5  x 12= x5+12 = x17 32  33 = = 35 76  713= = 719 Hal.: 3 Isi dengan Judul Halaman Terkait

4 The Division of Exponent Number
ap = ap-q, a = 0 aq Examples : : 52 = = 52 = 25 2. Hal.: 4 Isi dengan Judul Halaman Terkait

5 The Exponentiation of Exponent Number
(ap)2 = ap, ap, ap … ap… q factor = ap.q So (ap)q = ap.q Examples : 1. (52)3 = (5)2.3 = 56 = 15625 2. = 33 = 27 Hal.: 5 Isi dengan Judul Halaman Terkait

6 The Exponent of Double Multiplication or Numbers Greater
(ab)p = (ab) (ab) (ab)  . . . (ab) p factor (ab) = (a  b)  (a  b)  (a  b)   (a  b) p factor a and p factor b = (a  a  a  . . . a)  (b  b  b  . . . b) According to definition According to definition p faktor a p faktor a p factor a p faktor b p faktor b p factor b = ap  bp = apbp So (ab)p =apbp Examples : (3 7)5 = = 3575 = (2 2  3)5 = 25 25  35 = 210  35 = 21035 Hal.: 6 Isi dengan Judul Halaman Terkait

7 The Exponent Fraction Numbers
a  a  a  a  a  a …  a _______________________ = a  a  a...  a ap : aq = (p >q) a a  a … a p – q factor q factor number a = athe exponent ? Means  ap : aq = ap ‑ q = ap-q Examples : 36 : 34 = 36 ‑ 4 = 32 713 : 78 = 713-8 = 75 Hal.: 7 Isi dengan Judul Halaman Terkait

8 The Exponent Fraction Numbers
p factor p factor number a a  a  a  a  a  a …  a ap _______________________ ____ = = b  b  b  b  b  b …  b bp p factor number b ap So : ____ bp Hal.: 8 Isi dengan Judul Halaman Terkait

9 Isi dengan Judul Halaman Terkait
Zero Exponent Number If p, q are positive integer anumber and p = q then ap-q = a0 To determine the value of zero exponent number, look at this explanation below! a0 = ap-p ap = ap = 1 So, for every a R and a = 0 then we have a0 = 1 Hal.: 9 Isi dengan Judul Halaman Terkait

10 The Negative Exponent Number
ap = a0-p = a-p 1 a-p = ap a0 1 ap = ap So, for every a R, a = 0, and positive integer number then we have a-p = or ap = 1 a-p Examples : 1 5 = 2. Hal.: 10 Isi dengan Judul Halaman Terkait

11 Fraction Exponent Numbers
The exponent number of which is exponent by n can be rationalize as follows : (a ) p q q p q p q p q p q = a , a , a , … a as much as q a q. p q = ap = p (a ) q = is degined as exponent root at q from ap, then p = a q Hal.: 11 Isi dengan Judul Halaman Terkait

12 Fraction Exponent Numbers
Examples : 1. 2. 3. 4. Hal.: 12 Isi dengan Judul Halaman Terkait

13 The Properties of Exponent Numbers Operation
If a, b are real numbers, and p, q are integer numbers, then : ap  aq = ap+q ap : aq = ap-q ; a  0 (ap)q = apq (ab)p = ap bp . a-p = ; a  0. a0 = 1, a  0 b ; b  0 Hal.: 13 Isi dengan Judul Halaman Terkait

14 Isi dengan Judul Halaman Terkait
Roots 1. The Definition of Roots As we have discussed before, that Roots are numbers in the root symbol which cannot produce rational numbers Examples : Meanwhile : Because : 1, 2, and 8 are not irrational numbers Hal.: 14 Isi dengan Judul Halaman Terkait

15 Isi dengan Judul Halaman Terkait
Roots 2. Simplifying Roots Roots can be simplified by changing the number in the root into two numbers which one of them can be rooted and the other can not be rooted. Examples : 1. 2. Hal.: 15 Isi dengan Judul Halaman Terkait

16 Isi dengan Judul Halaman Terkait
Roots 3. Root Operation Operation base for a ≥ 0 and b ≥ 0 Addition and subtraction can be simplified if the roots are the same kind. Example : = = = Multiplication of roots using properties Examples : 1. 2. Hal.: 16 Isi dengan Judul Halaman Terkait

17 Isi dengan Judul Halaman Terkait
Roots Division of Roots (i) form Examples : 1. 2. Hal.: 17 Isi dengan Judul Halaman Terkait

18 Isi dengan Judul Halaman Terkait
Roots (ii) form Examples : 1. = = = = = 2. = = = = Hal.: 18 Isi dengan Judul Halaman Terkait

19 Isi dengan Judul Halaman Terkait
Roots (iii) form Example : = = = = Hal.: 19 Isi dengan Judul Halaman Terkait

20 Isi dengan Judul Halaman Terkait
Roots 4. Solving the exponent equation Properties used : ap = aq p = q Examples : Find the values of x that satisfy the following equations : 1. = 64 2. = Hal.: 20 Isi dengan Judul Halaman Terkait

21 Isi dengan Judul Halaman Terkait
Roots Answer : = 1. = 64 2. ↔ = 43 ↔ = ↔ 3x = 3 ↔ = ↔ x = 1 ↔ = ↔ = ↔ = ↔ = Hal.: 21 Isi dengan Judul Halaman Terkait

22 Isi dengan Judul Halaman Terkait
Logarithm Look at : ab = c ab = …. find the result of exponent …b = c find the exponent root of b from c a... = c find the exponent from a, so that the result is c = find the logarithm of base a from c number = alog c = … alog b = c  ac = b by a > 0 , a  1 and b > 0 a. Is base logarithm number b. Is number written in logarithm Hal.: 22 Isi dengan Judul Halaman Terkait

23 Isi dengan Judul Halaman Terkait
Logarithm The Properties If a > 0 , a  1 , m > 0 , n > 0 and x  R, then : alog ax = x alog (m.n) = alog m + alog n alog (m/n) = alog m - alog n alog mx = x. alog m alog m = If g > 0 , g  1 etc. an log b = alog b an log bm = alog b Hal.: 23 Isi dengan Judul Halaman Terkait

24 Isi dengan Judul Halaman Terkait
Logarithm Examples : 1. = 3 2. = 3 3. = 4. = = = 5 5. = = = 1 6. = = = 12 7. = = 8. = = = 1 9. = = = 6 Hal.: 24 Isi dengan Judul Halaman Terkait