Mathematics Class VIII Chapter 2 – Unit 4 Exponents.
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Transcript of Mathematics Class VIII Chapter 2 – Unit 4 Exponents.
Mathematics
Class VIIIChapter 2 – Unit 4
Exponents
Module Objectives
By the end of this chapter, you will be able to:
Understand the concept of an integral power to a non-zero
base
Write large numbers in exponential form
Know about the various laws of exponents and their use in
simplifying complicated expressions
Know about the validity of these laws of exponents for
algebraic variables.
INTRODUCTIONSuppose somebody asks you:HOW FAR IS THE SUN FROM THE EARTH? WHAT IS YOUR ANSWER……….?
A ray of light travels approximately at the speed of 2,99,792 km per second. It takes roughly 8 1/2min for a ray of light to reach earth starting from the sun
Hence the distance from the earth to the sun is about 15,29,00,000km. It takes 4.3light years at a speed of 2,99,792km per second.Which is to 4.3x365x24x60x60x299792km….. It will be difficult to read and comprehend. Here comes the help of exponential notation.
Exponent power base
5³ means 3factors of 5 or 5x5x5
353 3 means that is the exponential
form of t
Example:
he number
125 5 5
.125
So far this seems to be pretty
easy
Laws of Exponents
If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS
23 · 24 = 23+4 = 27 = 2·2·2·2·2·2·2 = 128
LAW#1 The Law of Multiplication
When the bases are different and the exponents of a and b are the same, we can multiply a and b first:a -n · b -n = (a · b) -n
Example:3-2 · 4-2 = (3·4)-2 = 12-2 = 1 / 122 = 1 / (12·12) = 1 / 144 = 0.0069444
When the bases and the exponents are different we have to calculate each exponent and then multiply:a -n · b -m
Example:3-2 · 4-3 = (1/9) · (1/64) = 1 / 576 = 0.0017361
512
2222 93636
So, I get it! When you multiply Powers, you add the exponents
When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS
LAW#2 The Law of Division
46 / 43 = 46-3 = 43 = 4·4·4 = 64
So, I get it!
When you divide powers, you subtract he exponents!
If you are raising a Power to an exponent, you multiply the exponents
LAW#3 Power of a Power
(23)4
You can simplify (23)4 = (23)(23)(23)(23) to the single power 212.
(35)2 = 310
(k-4)2 = k-8
(z3)y = z3y
-(62)10 = -(620)(3x108)3 = 33 x (108)3
= 27 x (108)3
= 27 x 1024
= 2.7 x 101 X 1024
= 2.7 x 101+24
= 2.7 x 1025(5t4)3 = 53 x (t4)3
= 53 x (t4x3) = 125 x t12
So when I take a power to a power, I
multiply the exponents
If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent.
LAW#4 Product of Exponents
(XY)3 = X3 x Y3
(ab)2 = ab × ab
4a2 × 3b2
[here the powers are same and the bases are different]
= (4a × 4a)×(3b × 3b)
= (4a × 3b)×(4a × 3b)
= 12ab × 12ab
= 122ab
So when I take a power of a product . I apply the
exponent to all factors of the product
n n nxy x y
If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator, each powered by the given exponent
LAW#5 Quotient Law of
Exponents
So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.
Law Examplex1 = x 61 = 6x0 = 1 70 = 1
x-1 = 1/x 4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5
xm/xn = xm-n x6/x2 = x6-2 = x4
(xm)n = xmn (x2)3 = x2×3 = x6
(xy)n = xnyn (xy)3 = x3y3
(x/y)n = xn/yn (x/y)2 = x2 / y2
x-n = 1/xn x-3 = 1/x3
Exponents are often used inarea problems to show the
areas are squared
A pool is rectangleLength(L) * Width(W) = AreaLength = 30 mWidth = 15 mArea = 30 x15 = 450sqm.
15m
30m
Exponents Are Often Used inVolume Problems to Show the
Centimeters Are Cubed
Length x width x height = volumeA box is a rectangleLength = 10 cm.Width = 10 cm.Height = 20 cm.Volume = 20x10x10 = 2,000 cm³
3
10
10
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