indices and surds (bilangan berpangkat dan bentuk …€¦ · (bilangan berpangkat dan bentuk akar)...
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1 |matematika kls X (wajib)
INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK AKAR)
Where n is a positive integer, na is defined as:
na a x a x a x ..... x a
n faktor
where a is called the base, and n, the index or exponent or power.
For example, 45 5 x 5 x 5 x 5
We shall restrict ourselves to positive bases (a > 0). Extending the definition to zero, negative and fractional
indices, we have the following results:
For a > 0 and positive integers p and q:
p qp
q
pp
p
po aaaaa
aa ,,1
,1
1
For example, 5 35
3
44
1
3
3 7755,8
1
2
12,12 ando
With these extended definitions, the following rules of indices hold for positive base, a, and any rational
indices, m and n.
m n m n
mm n
n
nm m n
a x a aa
aruler for same basea
a a
n n n
n n
n
a.b a x bruler for same index
a a
b b
2 |matematika kls X (wajib)
A number that cannot be expressed as a fraction of two integers is called an irrational number. Some
examples of irrational numbers are ,7,2 3 , etc. An irrational number involving a root is called a surd.
General rules involving surds:
n n np a q a p q . a ; n n n
p a q a p q a
n n na . b a.b ;
n
n
n
a a
bb
OVERVIEW
LAWS OF INDICES LAWS OF SURDS
a. am x an = am+n
b. am : an = am-n
c. (am)n = amn
d. a0 = 1
e. m
n mna a
f. a-n = n
1
a
g. an x bn = (ab)n
h. am : bm =
ma
b
i. m
n
n m
1a
a
j.
n na b
b a
k.
m m
n na b
b a
a. x . x x
b. x . y xy
c. a a x
xx
d. xyx x
ory yy
e. a x b x (a b) x
f. a x b x (a b) x
g. x y x y x y
h. 2 2a x b y a x b y a x b y
i. a x ya
x yx y
j. 2
x y x 2 xy y
k. 2
x y x 2 xy y
3 |matematika kls X (wajib)
Exercise:
1. Express each of the following in the surd form.
a. 3
1
x b. 4
5
6 c. 7
3
yx d. 3
5x y
e. 2
15
7
f.
21
32x . y
g.
1 1
3 4x . y
h. 3
5
3
12
.
yx i.
315
74x . y
2. Evaluate the following without the use of calculator:
a.
31
3 22
3
2
4 x 2
2
b.
1
1 32
2 4
2
5 x 5
3
c.
31
2 24
1
2
4 x 9
1
4
d.
1
1 52
3 6
1
2
5 x 5
4
e.
32
3
f. 2n 1 1 n n 1
9 x 3 : 27
3. If x y y z z x
a 0 , a a a
is equal ….
4. Simplify :
a.
1 1
2 2
x y
x y
b.
n 1 n 2
n 1 n 2
3 3
3 3
c.
n 1 2n 3
n 2n 1
4 2
4 2
5. If x 1
3 2
, then 2x
3
= ….
6. If x y
3 7 and 7 3 , find x . y
7. Express each of the following in the positive rational index form.
a. 5 b. 3 9 c. 4 243 d. 3
2
1 e. 6 x f. 3 3.yx g. 2yx
h. n px i. n qp yx . j.
3 2
4 2
y
x
k. 3 23 . yx
4 |matematika kls X (wajib)
8. Evaluate.
a. 64 3
2
b. 125 3
2
c. 625 4
3
d. 81 4
3
e. 32
77
f. 3
1
125
1
g.
4
22
1
h.
5
4
32
1
i. 243 3
2
9. Simplify each of the following, giving your answer in the surd form.
a. 2
9
3 .
xx b. 4 33 2 . xx c.
3
1
3 2
y
x d.
ba
ba
ba.
.
. 2
5
3
e. 8 3 22 xxx f.
3 4 3 22 xxx g.
4
3
4
12
6
1
4
1
3
1
2
1
:.
x
y
x
yyx
h. 3
1
3 4 1234
1
aaaa i.
33
3 4
3
13
3
133.
1
13
xx
xxx
x
x
j.
3
113
22
3
1
3 23
2
3
2
3
11
.
cb
a
a
c
ca
ba k.
5
2
5
3
24
3
3
2
4
2
8
21627
l. 3522
2
1
2
1
2
1.
2
1.25,0.125,0.2.4
m.
33
11
36
1
2
114 9 823
2
..283.2..21.64 baba
5 |matematika kls X (wajib)
10. Simplify each of the following surds.
a. 272523 b. 2
15018 c. 28
2
1
4
3172175
d. 4444 1623281
22.3 e. 3333
9
11923000
8
3.
5
1
f. 32
32
33 8.
2
127
a
b
b
aabab g. 3333
9
13000
8
3.5375
h. 55.4.22.3 i. 21.632 j. 7.65.27.55.3
k. 3333 64.32 l. baccba ... m. 75.3.75.3
n. 51.51 o. qpqpqp .. 4444
p. 2.. abba q. 25.26 r. 25.27.3
s. 357.2 t. 3.2..5 aba u. 22.33.26
v. 22 ........ babababaabba w. 333 25.45.1236.5..26
Rationalisation of the denominator:
The general form of conjugate surds are ba and ba . The product of a pair of conjugate surds is
always a rational number.
11. By rationalising the denominators, simplify:
a. 6
3607248 b.
3
211524 c.
3
43
3
222.5 d.
6
34
2
61218
e. 53
4
f.
3.24
3.2
g.
210
35.7
h.
35
35.2
6 |matematika kls X (wajib)
i. 7.5.3
7.5.3
21
21
j.
yxxy
yxyx
..
..
k.
23
2.23.3
l.
33 ba
ba
m. 1
13
a
a n.
321
1
o.
73.25
35
p.
22322
21
q. 3 233 2 . bbaa
ba
r.
177
2
33 2 s.
333
33
469
23
baabba .2 and babaabba ;.2
12. Express in a b form,
a. 10.27 b. 15.28 c. 21.210 d. 78.219
e. 110.221 f. 130223 g. 2.46 h. 6.411
i. 5.614 j. 3.1452 k. 2.1027 l. 2.3055
m. 74 n. 32 o. 5.37 p. 70421
41
q. 65.327
13. a. 7474 b. 537537 c. 10.2910.29
d. 6.386.38 e. 53.5353.53 f. 33 13251325
14. Evaluate: 32.2
2
2.27
2.....
23
2
32
2
21
2
.
7 |matematika kls X (wajib)
Advanced Exercise:
1. 12...12.12.12 512842
2. 13
1
13
1
13
1...
13
1
13
1
13
110009999989989991000
.
3. a. 481353.2 b. 4 13136497
4. a. 60402410 b. 7
23.8.3114
5. 5210.285210.28
6. 3222322232232
7. If x = 3819 , find 158
2318262
234
xx
xxxx.
8. 33
3
1.
3
8
3
1.
3
8
aaa
aaa .
9. Jika 123;123;123 zyx , maka
x2 + y
2 + z
2 + xy + yz + zx = ....
10. Jika x2 + 12x + 1 = 0, maka nilai dari
4
4 1
xx = ….
11. Rasionalkan penyebut:
a. 7523
5213515
b.
333
3
2427
116
12. Nilai x yang memenuhi 33
xxx adalah ….
8 |matematika kls X (wajib)
13. Kurva xy 111 berpotongan dengan garis y = x di titik (a, b), maka nilai
a2 – b = ....
14. Nilai dari ...151413121 = ….
15. Bentuk sederhana dari:
a. 116116 b. 33 5252
LATIHAN BENTUK PANGKAT DAN AKAR
I. Jadikan bentuk √a + √b :
1. 6 2 5 2. 13 4 10 3. 10 2 21 4. 7 40
5. 6 2 4 2 3 6. 4 7 7. 6 2 5 8. 19 4 15
9. 12 2 35 10. 20 2 91 11. 7 4 3 12. 123 22 2
13. 80 28 10 14. 152 30 15 15. 7 3 5 16. 5 2 29 3 3
17. 4 57 24 3 18. 2 9 4 2
2 1
19. 32 10 7
20. 117 36 10 21. 28 5 12 22. 12
4 2 2
23. 2 3. 2 2 3 . 2 2 2 3 . 2 2 2 3 . . .
9 |matematika kls X (wajib)
II. SEDERHANAKAN/HITUNGLAH :
1.)
514 1 6 232
16 5 3 3
( )a b a b c
a b c
. . .
2.)
2
22 15 10
5 3
. . .
3.) 12 2
1 1 1 1 22
2 2 2 2
1 2 21
:
a a a
aa aa a a a
. . .
4.)
11 1
1 1
x y xy
x y
. . .
5.)24 2 3
752 2 3
. . .
6.) 104 61
0,25 1,44 x10 22,5 10 243 1527
. . .
III. RASIONALKAN :
1.) 1
7 4 3
2.)
1
1 2 3
3.
33 3
7
16 12 9
IV. PILIHAN GANDA
1. Diketahui : 6x + y = 36 dan 6x + 5y = 216, maka harga x = . . .
a. 14
b. 34
c. 54
d. 32
e. ) 74
2. Jika xy = 7, maka nilai
2
2
( )
( )
2
2
x y
x y
. . .
a. 22 b. 27 c. 214 d). 228 e. 2196
3. Jika 3x – 3x – 3 = 78√3; maka nilai x = . . .
a. 3√3 b. 32
√3 c. 81√3 d). 92
e. 94
10 |matematika kls X (wajib)
4. Jika 12( )x xa e e dan 1
2( )x xb e e maka nilai
22 2a b . . .
a. 2xe b. 2xe c. 2 2xe e d) 1 e. 0
5. 3 32 49 2169 3 8 12 64 8 50
13 16 5
a. – 29 b. – 11 c. 5 d. 17 e) 24
6. Nilai x yang memenuhi persamaan : 2 2
3 7
13
27
x
x
adalah :
a) 2,5 b. 2 c. 1 d. – 2,5 e. – 1,25
7. Nilai x yang memenuhi : 2 4
84 2
x
x adalah :
a. 15
2 b.
13
2 c.
11
2 d.
9
2 e.
7
2
“Saya tidak pernah meminta agar Tuhan menjadikan hidup ini mudah. Saya hanya meminta agar Ia
menjadikan saya kuat.”
11 |matematika kls X (wajib)
LOGARITHMS
If a number (b) is expressed as the exponent c of a number (a), i.e. , cb=a a>0, a 1 , we say that c is
the logarithm of b to the base a. We write this as a logb=c , sometimes as alog b=c .
In general: c
b=a alogb = c , a>0, a≠0
For example, 2100log2100log10100 102 or 38
1log2
8
1 23
Exercise:
1. Convert the following to logarithm form:
a. 8134 b. 49
17 2 c. rpq
2. Convert the following to exponential form:
a. 532log2 b. 29log3 c. rqp log
3. Find the value of each of the following:
a. 64log2 b. 4log2
1
c. 1log3 d. 7log7
e. 25,0log8 f. )9log(3 g. 9log81 h. 32log22
4. Find x: a. 5
1164log x b.
5log5
15log2
x
x
= 1
Note: a. logarithms of a positive number may be negative
b. logarithms of 1 to any base is 0 i.e. 01log a
c. logarithms of a number to base of the same number is 1 i.e. 1log aa
d. logarithms of negative numbers are not defined, for example )4log(2
e. the base of a logarithm cannot be negative, 0 or 1. Can you think of why this is so ?
Laws of logarithms:
1. ba ba
log
2. n
mna
babm
log
3. cbcb aaa .logloglog
4. c
bcb aaa logloglog
5. bnb ana loglog
6. bb a
mnnam
loglog
12 |matematika kls X (wajib)
5. Prove laws of logarithms no. 1 – 6.
6. Find the value of each the following:
a. 25log4
4 b. 2log5
5 c. 4log3
9 d. 6log125
25 e. 8log2
8
f. 5log27
3 g. 10log2
4 h. 7log625
55 i. 2
19 log381 j. 6log
41
8
2
7. Simplify and evaluate:
a. log 25 + log 4 b. 25log200log 44 c. 250log8log5log 5
2
5
2
5
2
d. 2 log 2 + 2 log 3 + 3
1 log 5 + log 7 – log 9 + log 10 + log 3 25.5 -
2
1 log 49
e. 144log.14log10log9log.7log5log 3
21333
3133
8. Expand to a single logarithm:
a.
2
3 .log
z
yx b.
53.log
z
yx c.
22
22
logyx
yx d. yxx .log 234
9. Given that log 2 = 0,3010 dan log 3 = 0,4771, find
a. log 0,002 b. log 3000 c. log 6000 d. log 15 e. log 3
4 f. log
24
5 g. log 3 6.3
10. Evaluate:
a. 2log
2log4log
3
3
9
2
627 b.
8log54log.516log2log.3
512log.8
c.
4,0log
5log2log 22
11. a. If my
xa
log , find 6 22 :log yxa .
b. If nxyb 3 :log , find 5 3:log yxb
13 |matematika kls X (wajib)
Laws of logarithms:
7. a
bb
p
pa
log
loglog
8. ccb aba logloglog
9. naa bbn
loglog
10. a
bb
a
log
1log
12. Prove laws of logarithms no. 7 – 10.
13. Simplify:
a. 51352 log64log27log b.
43
2
loglog
log
bd
c
ac
b
a
14. Evaluate:
a. 81log
1
81log
118
2
1 b.
5log2
125log
25log
1
2
1
4
1
15. Simplify: 118log8log3log10log 252555
525
.
16. a. If p5log27 , find 55log243 .
b. Given p8log5 , find 125,0log2,0 .
c. Given a27log25 , find 5log9 .
d. If m27log16 , find 8log3 .
e. Given a5log4 , find 25,1log1,0 .
17. Given a3log4 , express the following in a:
a. 3log2 b. 81log8 c. 9116 log
18. Given ba pp 30log,5log and cp 21log . Express the following in a, b or c.
a. 211logp b. 10logp c. 36logp
19. Given m30log6 and n20log6 . Express 3log6 in m or n.
14 |matematika kls X (wajib)
20. Simplify 9log9log
3log25
.
21. If a5log3 and b8log25 , find 750log15 .
22. For a, b and M are greater than 1, and xaM Mabb
loglog , find x.
23. Prove : bcabccba
abab
loglog .
24. Given
a
bc
a
cb
a b
b
c
c
ba log
log
log
log
log
11
2
. Prove a + b = c.
25. Given ayx 2log and by
xlog . Find xy log .
26. Given ba 7log,5log 22 and c5log3 . Express 98log48 in a, b or c.
27. Evaluate: 9log92
2
342
1
216log.3log5log10log
20log
15log
15log
.
28. Evaluate:
3 2 3 2 4 936
2 33 3
log 36 log 4 log125 log125. log5 5
log25. log25log12 . 144
.
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