Mathematics  Class VIII  Chapter 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.

INTRODUCTION Suppose 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

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 2 3 · 2 4 = = 2 7 = 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 = = 1 / 12 2 = 1 / (12·12) = 1 / 144 = 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 =

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 4 6 / 4 3 = = 4 3 = 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 (2 3 ) 4 You can simplify (2 3 ) 4 = (2 3 )(2 3 )(2 3 )(2 3 ) to the single power 2 12.

(3 5 ) 2 = 3 10 (k -4 ) 2 = k -8 (z 3 ) y = z 3y -(6 2 ) 10 = -(6 20 ) (3x10 8 ) 3 = 3 3 x (10 8 ) 3 = 27 x (10 8 ) 3 = 27 x = 2.7 x 10 1 X = 2.7 x = 2.7 x (5t 4 ) 3 = 5 3 x (t 4 ) 3 = 5 3 x (t 4x3 ) = 125 x t 12

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 = X 3 x Y 3

(ab) 2 = ab × ab 4a 2 × 3b 2 [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

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.

LawExample x1 = x61 = 6 x 0 = 17 0 = 1 x -1 = 1/x4 -1 = 1/4 x m x n = x m+n x 2 x 3 = x 2+3 = x 5 x m /x n = x m-n x 6 /x 2 = x 6-2 = x 4 (x m ) n = x mn (x 2 ) 3 = x 2×3 = x 6 (xy) n = x n y n (xy) 3 = x 3 y 3 (x/y) n = x n /y n (x/y) 2 = x 2 / y 2 x -n = 1/x n x -3 = 1/x 3

Exponents are often used in area problems to show the areas are squared A pool is rectangle Length(L) * Width(W) = Area Length = 30 m Width = 15 m Area = 30 x15 = 450sqm. 15m 30m

Exponents Are Often Used in Volume Problems to Show the Centimeters Are Cubed Length x width x height = volume A box is a rectangle Length = 10 cm. Width = 10 cm. Height = 20 cm. Volume = 20x10x10 = 2,000 cm³