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Algebraic Terms 2016
II. Algebraic Forms and Processes
PART I: Powers, Roots, and Logarithms
Words
Pronunciation
Powers
Indices
Base
/'paʊə(r)z/
/’næt∫ral ’nΛmbə(r)z/
Basis
/'beɪsəs/
Exponent
Roots
Radicals
Radical sign
Radicand
Perfect square
Binomial
Conjugate
Logarithm
/ b e ɪs /
Indonesian
Bilangan Berpangkat
Bilangan Asli
Bilangan Pokok
Bilangan Pokok
/rædɪklz/
Pangkat
Bentuk Akar
Bentuk Akar
/rædɪkən/
Bilangan yang diakarkan
/ɪk'spəʊnənt/
/ru:tz/
/rædɪklz saɪn/
Tanda Akar
/’pɜ:fɪkt skweə(r)/
Kuadrat Sempurna
/'kɒndƷʊgeɪt/
Sekawan
/baɪ'nəʊmɪəl/
/'lɒgəriðəm/
Binomial
Logaritma
Example:
•
•
•
•
•
•
•
•
Powers or indices is used when we want to multiply a number by itself
several times.
In xn term, x is called base or basis and n is called exponent.
Root is inversion of exponentiation.
n
a is called radical expression (or radical form) because it contains a
root.
In 3 a + b , a+b is the radicand, and 3 is the index.
A number is said perfect square if its roots are integers.
Numbers such as 9, 16, 36, and 100 are perfect squares, but 12 and 20
are not.
The square root is in simplest form if the radicand does not contain
perfect squares other than 1, no fraction is contained in radicand, and
•
•
no radicals appear in the denominator of a fraction.
A radical and a number is called a binomial.
2 + 6 is a binomial and its conjugate is 2 − 6
1 Algebraic Terms
How to Say Powers, Roots, and Logarithms
•
x2
•
x3
•
xn
•
•
•
•
•
(x+y)2
•
•
•
•
x
5
3
y
n
z
•
•
•
x2y3
nlog
x
ln 2
5log2
25
•
•
•
•
•
•
•
•
Algebraic Terms 2016
x squared
x cubed
/'skweə(r)d/
/kju:bd/
x to the power of n
x to the n-th power
x to the n
x to the n-th
x u p pe r n
x raised by n
/'Λpə(r)/
/reizd/
x plus y all squared
bracket x plus y bracket closed squared
/'brækit/
x plus y in bracket squared
square root of x
root of x
cube root of y
n-th root of z
fifth root of (pause) x squared times y cubed
fifth root of x squared times y cubed in bracket
log x to the base of n
log base n of x
natural log of two
“L N” of two
log squared of twenty-five to the base of five
log base five of twenty-five all squared
2 Algebraic Terms
/lɒg/
Algebraic Terms 2016
More Examples
2log
(x+y)+2 2log 4x >4
x2 +
1
= 1
x
3 +9
x
> 27
x −1
9x − 1 < 2
log base two of x plus y in bracket plus two times
log base two of four x’s is greater than f our
x squared plus (pause) one over root of x equals
one
three upper x plus (pause) nine upper x minus
one (pause) is more than twenty-seven
nine to the x (pause) minus one is less than two
Practice
1. Read out the following terms.
a. 26
2
b.
3
3
c. x5 : x2=x3
d. (3ab)4
x
e.
3y
3
f. (9x)0
4x 4
g.
h.
4
m3n8 =m3/4n2
i.
5
a 3 =a3/5
j.
3
8x 6 y 9
k.
x2 + y2
l. a xlog b
m. log a2
n. 2log (1/6)
o.
5log
(x2+y)
p. (nlog x)2
q. 6log2 22 – 6log x2 -1
3 Algebraic Terms
2. Read these expressions and simplify them.
a. 53×513
(2 )
Algebraic Terms 2016
b. 814 : 811
c.
4
3
x7
d. 3
x
2
e. 2431/5
f. -4-2
g. 1251/3
h. (-5)-1
i. 3-3
j.
72
k.
234
l.
5
2+ 3
m.
3
6− 2
PART II: Algebraic Process
Words
Expand
Simplify
Factorize
Cancel
Substitute
Side
Left-hand side
Right-hand side
Collect
Eliminate
Pronunciation
/ ɪk’spænd/
/’sɪmplɪfaɪ/
/’fæktəraɪz/
/’kænsl/
/’sΛbstɪtju:t/
/saɪd/
/left hænd saɪd/
/raɪt hænd saɪd/
/kə’ləkt/
/ɪ’lɪmɪneɪt/
4 Algebraic Terms
Indonesian
Uraikan/Jabarkan
Sederhanakan
Faktorkan
Coret/Hapus/Batalkan
Substitusi
Ruas
Ruas kiri
Ruas kanan
Kumpulkan
Eliminasi
Examples:
1. Expand (x-3)(x+2) into x2-x-6.
Algebraic Terms 2016
2. Simplify the left hand side (LHS) of equation (2x+2)(xy+y) =5y into
2y(x+1)=5y
3. Factorize x3-2x2+3x-2 into (x-1)(x+1)(x-2)
4. Cancel (x+1) from 2(x+1)/(x+1) to get 2
5. Multiply both side of equation ½x = 4 with 2 to get x=8
6. Substitute y=4 into equation 2x+y=12 and we have 2x+4=12
7. Collect (x+2) from (x+2)3-2(x+2)(x+1) to get (x+2)[(x+2)2-2(x+1)]
Example
x x-1
Find x that satisfy equation 3 -3
=162.
Answer
x x
First, we multiply both side with 3, and we have 3.3 -3 =486.
x
x
Then, we collect 3 , and we get 3 (3-1)=486, which can be simplified
x
into 2.3 =486.
x
Divide both side by 2, we get 3 =243.
5
We know that number on the right hand side (RHS) equals 3 , so we can
x 5
write 3 =3 .
According to the rule of powers, x must be equal to 5.
Exercise
Explain the process to find the solution of problems below.
Find x that satisfies each equation, inequality, or system of equations.
1. 1 – 4x ≤ x + 11
2.
3.
3
5
4 - 2y
y- =
2
3
5
x +=
y 2
2 x − y = −5
5 Algebraic Terms
II. Algebraic Forms and Processes
PART I: Powers, Roots, and Logarithms
Words
Pronunciation
Powers
Indices
Base
/'paʊə(r)z/
/’næt∫ral ’nΛmbə(r)z/
Basis
/'beɪsəs/
Exponent
Roots
Radicals
Radical sign
Radicand
Perfect square
Binomial
Conjugate
Logarithm
/ b e ɪs /
Indonesian
Bilangan Berpangkat
Bilangan Asli
Bilangan Pokok
Bilangan Pokok
/rædɪklz/
Pangkat
Bentuk Akar
Bentuk Akar
/rædɪkən/
Bilangan yang diakarkan
/ɪk'spəʊnənt/
/ru:tz/
/rædɪklz saɪn/
Tanda Akar
/’pɜ:fɪkt skweə(r)/
Kuadrat Sempurna
/'kɒndƷʊgeɪt/
Sekawan
/baɪ'nəʊmɪəl/
/'lɒgəriðəm/
Binomial
Logaritma
Example:
•
•
•
•
•
•
•
•
Powers or indices is used when we want to multiply a number by itself
several times.
In xn term, x is called base or basis and n is called exponent.
Root is inversion of exponentiation.
n
a is called radical expression (or radical form) because it contains a
root.
In 3 a + b , a+b is the radicand, and 3 is the index.
A number is said perfect square if its roots are integers.
Numbers such as 9, 16, 36, and 100 are perfect squares, but 12 and 20
are not.
The square root is in simplest form if the radicand does not contain
perfect squares other than 1, no fraction is contained in radicand, and
•
•
no radicals appear in the denominator of a fraction.
A radical and a number is called a binomial.
2 + 6 is a binomial and its conjugate is 2 − 6
1 Algebraic Terms
How to Say Powers, Roots, and Logarithms
•
x2
•
x3
•
xn
•
•
•
•
•
(x+y)2
•
•
•
•
x
5
3
y
n
z
•
•
•
x2y3
nlog
x
ln 2
5log2
25
•
•
•
•
•
•
•
•
Algebraic Terms 2016
x squared
x cubed
/'skweə(r)d/
/kju:bd/
x to the power of n
x to the n-th power
x to the n
x to the n-th
x u p pe r n
x raised by n
/'Λpə(r)/
/reizd/
x plus y all squared
bracket x plus y bracket closed squared
/'brækit/
x plus y in bracket squared
square root of x
root of x
cube root of y
n-th root of z
fifth root of (pause) x squared times y cubed
fifth root of x squared times y cubed in bracket
log x to the base of n
log base n of x
natural log of two
“L N” of two
log squared of twenty-five to the base of five
log base five of twenty-five all squared
2 Algebraic Terms
/lɒg/
Algebraic Terms 2016
More Examples
2log
(x+y)+2 2log 4x >4
x2 +
1
= 1
x
3 +9
x
> 27
x −1
9x − 1 < 2
log base two of x plus y in bracket plus two times
log base two of four x’s is greater than f our
x squared plus (pause) one over root of x equals
one
three upper x plus (pause) nine upper x minus
one (pause) is more than twenty-seven
nine to the x (pause) minus one is less than two
Practice
1. Read out the following terms.
a. 26
2
b.
3
3
c. x5 : x2=x3
d. (3ab)4
x
e.
3y
3
f. (9x)0
4x 4
g.
h.
4
m3n8 =m3/4n2
i.
5
a 3 =a3/5
j.
3
8x 6 y 9
k.
x2 + y2
l. a xlog b
m. log a2
n. 2log (1/6)
o.
5log
(x2+y)
p. (nlog x)2
q. 6log2 22 – 6log x2 -1
3 Algebraic Terms
2. Read these expressions and simplify them.
a. 53×513
(2 )
Algebraic Terms 2016
b. 814 : 811
c.
4
3
x7
d. 3
x
2
e. 2431/5
f. -4-2
g. 1251/3
h. (-5)-1
i. 3-3
j.
72
k.
234
l.
5
2+ 3
m.
3
6− 2
PART II: Algebraic Process
Words
Expand
Simplify
Factorize
Cancel
Substitute
Side
Left-hand side
Right-hand side
Collect
Eliminate
Pronunciation
/ ɪk’spænd/
/’sɪmplɪfaɪ/
/’fæktəraɪz/
/’kænsl/
/’sΛbstɪtju:t/
/saɪd/
/left hænd saɪd/
/raɪt hænd saɪd/
/kə’ləkt/
/ɪ’lɪmɪneɪt/
4 Algebraic Terms
Indonesian
Uraikan/Jabarkan
Sederhanakan
Faktorkan
Coret/Hapus/Batalkan
Substitusi
Ruas
Ruas kiri
Ruas kanan
Kumpulkan
Eliminasi
Examples:
1. Expand (x-3)(x+2) into x2-x-6.
Algebraic Terms 2016
2. Simplify the left hand side (LHS) of equation (2x+2)(xy+y) =5y into
2y(x+1)=5y
3. Factorize x3-2x2+3x-2 into (x-1)(x+1)(x-2)
4. Cancel (x+1) from 2(x+1)/(x+1) to get 2
5. Multiply both side of equation ½x = 4 with 2 to get x=8
6. Substitute y=4 into equation 2x+y=12 and we have 2x+4=12
7. Collect (x+2) from (x+2)3-2(x+2)(x+1) to get (x+2)[(x+2)2-2(x+1)]
Example
x x-1
Find x that satisfy equation 3 -3
=162.
Answer
x x
First, we multiply both side with 3, and we have 3.3 -3 =486.
x
x
Then, we collect 3 , and we get 3 (3-1)=486, which can be simplified
x
into 2.3 =486.
x
Divide both side by 2, we get 3 =243.
5
We know that number on the right hand side (RHS) equals 3 , so we can
x 5
write 3 =3 .
According to the rule of powers, x must be equal to 5.
Exercise
Explain the process to find the solution of problems below.
Find x that satisfies each equation, inequality, or system of equations.
1. 1 – 4x ≤ x + 11
2.
3.
3
5
4 - 2y
y- =
2
3
5
x +=
y 2
2 x − y = −5
5 Algebraic Terms