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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