How Large Is One Zillion? - Diwa Learning Systems

How Large Is One Zillion?
By Divina Gracia T. Bandong
How much is 1 000 000 000? Do you have an idea? Suppose you could spend 1 000.00 a day
every day. How long would it take you to spend 1 000 000 000? Assume that each year has 365
days. The answer should be approximately 2 740 years.
Name Value
million 106 billion 109 trillion 1012 quadrillion 1015 quintillion 1018 sextillion 1021 septillion 1024 octillion 1027 nonillion 1030 decillion 1033 undecillion 1036 duodecillion 1039 tredecillion 1042 quatuordecillion 1045 quindecillion 1048 sexdecillion 1051 septendecillion 1054 octodecillion 1057 novemdecillion 1060 vigindecillion 1063 googol 10100 TATSULOK Second Year Vol. 11 No. 3
Did you know that a billion for British means a
trillion for Americans? What Americans call a
billion, the British call a thousand million. This
is because the American system of naming large
numbers is based on powers of 10 that are multiples
of three while that of the British system is based
on powers of 10 that are multiples of 6. We may
not agree with the names for large numbers, but their
values should be clear.
One million is a thousand thousand and is written
as 1 000 000. One billion, on the other hand, is one thousand million,
written as one followed by nine zeros. Million, billion—what’s next? How
about a number written as 1 followed by 12 zeros, or a number one followed
by 60 zeros? Have you imagined writing a number 1 followed by a lot of
zeros? It would be very inconvenient. Hence, exponents are devised to
easily handle big numbers. On the left are the names of big numbers and
their values, which are written using exponents. This naming is based on the
American system.
Is there such a thing as one zillion? What is its value? Zillion is just
a term that is used to denote a number or quantity so huge that it cannot be
counted nor determined. It is modeled on million and billion, with Z being
the last letter in the English alphabet,
representing the last in a series.
Rules of Exponents
Exponent is a number or variable
placed to the upper right of a number
or mathematical expression that indicates the number of times
the number or expression is to be multiplied by itself. The word
exponent comes from the Latin exponent, the present participle stem
of exponere, which means “to expound.”
When factors of a product are the same, the product is called a
power of the repeated factor. We denote it as follows:
xn googolplex 10
= x • x • x • x
n factors of x
googol
where n is a positive integer and xn is the nth power of x. In the
symbol xn , x is called the base and n the exponent of the power.
When dealing with exponential expressions, certain rules have
to be followed (see sidebar). Make sure you know them by heart and
how to apply them.
➊ x1 = x
➋ xm · xn = xm+n ➌ xm
x n = x m–n , x ≠ 0
➍ (xy) n = x n y n
➎ ( x
n
y ) = xn
yn , y ≠ 0
➏ (xm ) n
= xmn ➐ x0 = 1, x ≠ 0
➑ x –n = 1
xn , x ≠ 0
1
➒
x –n = xn , x ≠ 0
Caution!
In general, (x + y) n ≠ xn + yn .
e-Pages
2

The preceding rules of exponents are used when we want to simplify an algebraic expression. A
combination of these rules may be applied in simplifying a single expression. How do we know if the
expression is already in simplified form? Bear these in mind.
➊ There should be no grouping symbols. ➍ Each factor has a single positive integral exponent.
➋ There should be no negative or zero exponents. ➎ Each factor or variable appears only once in the expression.
➌ The fractional coefficient is in lowest term.
=
=
=
22 x2 2
)
3 2
=
2 ___________________
xn
yn (22 ) 2 ( x2 )
9
24 =
4 x
(x
___________________
9
4 16 x
___________________
m ) n
= xmn Exercise 1
( 6x
8x3 2
2 · 3 · x
) = ( 23 x3 2
) ___________________
= (21–3 · 3 · x1–3 ) 2
___________________
= (2 –2 · 3x –2 ) 2
___________________
= ( 3
3(27a
___________________
2b –3 ) –2
(9a4b3c0 )
= 3(33a2b –3 ) –2
(32a4b3c0 ) ____________________________
= 3(3 –6a –4b6 ) (32a4b3 ) ____________________________
= 31+(–6)+2a –4+4b6+3 ____________________________
= 3 –3a0b9 Can you justify each step in the following simplification? Write the procedure used or the rule/s
of exponents applied in each step.
x
____________________________
m
xn = xm–n , x ≠ 0
x –n = 1
xn (x
, x ≠ 0
m ) n
= xmn and x0 = 1, x ≠ 0
xm · xn = xm+n ( x
y ) n
Exponents are used in writing a number in scientific notation. Scientific notation is a shorthand
way of writing very large or very small numbers. As the name suggests, scientific notation is especially
useful to scientists in writing or reading extremely large or small numbers since writing numbers in
scientific notation allows them to leave out a number of zeros while retaining some indication that the
zeros exist.
A number x is written in scientific notation as a number between one and 10 multiplied by a specific
power of 10. In symbols,
x = a × 10 n where 1 < a < 10 and n is an integer
How do you express a number in scientific notation? Simply move the decimal
point up to a number between one and 10. Then, count the number of places in
which you moved the decimal point. This becomes the power of 10 in the scientific
notation. A large number is written in scientific notation with a positive power of 10
while a very small number takes a negative power of 10. This means that when you
move the decimal point to the left, the exponent of 10 is positive, while a negative
exponent is used if you move the decimal point to the right. Look at this example.
Example:
Scientists have defined the speed of light in a vacuum to be approximately
300 000 000 meters per second. This is written in scientific notation as:
300 000 000 = 3 × 10 8
8 places
TATSULOK Second Year Vol. 11 No. 3 e-Pages
b 9
3 3
x –n = 1
____________________________
xn , x ≠ 0 and x0 = 1, x ≠ 0
3

ACROSS
2. (a8 ) 20
· a8 = a ???
x3y –2
x –6 y2 = x?
y ?
4.
(2z
6.
–3 ) –2
z8 (–z5 ) 6 = 1
4z ??
1
7.
(3 –2 2 = ??
)
8. (2 · 4 · 8 · 16) 2 = 2 ??
3(2a)
9.
6
(2b) 3 = ??a6
b3 12. [3(2xy) 2 ] 2
= ???x 4 y4 15. 9x+2
= ??
2x 3
1
17.
0 + 20 + 30 + 40 + 50 (–2) –2 = ??
18. (x2yz0 ) 2 · y = x ? y ?
(2x
20.
2 )(5x –3 )
(3x) –1 = ??
21. 0.00000000060 = 6 × 10 ?
22. (ab2c3 ) 2 (a –2b –1c) –2
= a ? b ? c ?
DOWN
1. 4.1 × 10 –6 = ?????????
2. 5(a 2 – b 2 ) = ??? if a = 5 and b = 1
3. 820 000 000 = ??? × 108 4 · 73
4. = ??
14
5. 4.1 × 108 = ?????????
6. (xy2 ) 2
= ?? if x = –2 and y = –3
10. 3(2z8 ) 2
· 4z -16
= ??
Exercise 2
Let’s see how well you work with exponents. Complete the following number puzzle. Solve the
given problems below. A question mark represents a digit or a decimal point. A single digit or a decimal
point occupies one box in the grid. The digits/decimal point are written on the grid in the same order that
they are written in the expression. If the missing digits are written from top to bottom, write them in the
grid from left to right or from top to bottom.
1 2 3 4 5
6 7
8 9 10
12
11
13 14
15 16 17
18 19 20
21 22
11. [ 49(3u2v) 3
63 ]
(3x)
13.
2
(2y) –1 = ??x2y 14. 3 × 102 = ???
2
= ???u 12 v 6
16. 0.00016 = ??? × 10 –4
17. a –1 (b –3 )c · (ab 2 c) 3 = a ? b ? c ?
19.
9 × 10 5
3 × 10 4 = ??
TATSULOK Second Year Vol. 11 No. 3 e-Pages
4

Answers to Exercises
How Large Is One Zillion?
Exercise 1
A. Factoring
xm xn = xm–n , x ≠ 0
Addition of Integers
x –n = 1
xn , x ≠ 0
( x
y ) n
= xn
y n
(xm ) n
= xmn Expanding the power
B. Expressing numbers in terms of
exponents
Exercise 2
(xm ) n
= xmn and x0 = 1, x ≠ 0
x m · x n = x m+n
Addition of integers
x –n = 1
x n , x ≠ 0 and x 0 = 1, x ≠ 0
1 2 3 4 5
0 1 6 8 9 4
6 7
. 3 2 . 8 1
8 9 10
0 2 0 2 4 0
12
11
0 4 4 8 0
0 1 4 4 0
13 14
0 1 1 3 0
15 16 17
0 8 1 2 0 0
18 19 20
4 3 . 3 0 0
21 22
1 0 6 4 4 0
Factor In
Answer to the Preliminary Problem:
0, one of the factors is (x – x) = 0
TATSULOK Second Year Vol. 11 No. 3 e-Pages
I.
1. Factor = 3ab (a2 – 3ab + 9b) 2
2. Factor = (m – n)(3m – 1)
3. Factor = 4(u – 1)(2 – v)
II.
(1) To factor x2 + 5x + 6, I need two numbers that add
up to 5 and have a product of 6.
Thus, x2 + 5x + 6 = (x + 5)(x + 1).
(2) To factor x2 – 2x – 3, I need two numbers that add
up to –2 and have a product of –3.
Thus, x2 – 2x – 3 = (x – 3)(x + 1).
(3) To factor x2 – 8x + 15, I need two numbers that
add up to –8 and have a product of 15.
Thus, x2 – 8x + 15 = (x – 3)(x – 5).
III.
1. (x + 7) and (x + 4)
2. (x + 3) 2
3. (x + 4) and (x – 1)
4. (4x – 1) and (3x – 4)
5. a b
c
d
8