1 |matematika kls X (wajib) INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK AKAR) Where n is a positive integer, n a is defined as: n a axaxax ..... xa n faktor where a is called the base, and n, the index or exponent or power. For example, 4 5 5x5x5x5 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 q p q p p p p o a a a a a a a , , 1 , 1 1 For example, 5 3 5 3 4 4 1 3 3 7 7 5 5 , 8 1 2 1 2 , 1 2 and o With these extended definitions, the following rules of indices hold for positive base, a, and any rational indices, m and n. m n mn m mn n n m mn a xa a a a ruler for same base a a a n n n n n n a.b a xb ruler for same index a a b b
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INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK …€¦ · (BILANGAN BERPANGKAT DAN BENTUK AKAR) Where n is a positive integer, an is defined as: a a x a x a x ..... x an nfaktor
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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