1 INDICES, SURDS AND LOGARITHM

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Lydia Valensia X Grade Mathematics is the way you think

GRADE X

STANDARD COMPETENCE: 1. Solving problems related to indices, surds and logarithms BASIC COMPETENCE:

Using laws of indices, surds and logarithms. Doing the algebraic manipulation in computation related to indices, surds and logarithms. (Integrated with E- SETS )

Hello, my students…. Today, we are going to learn about indices, surds and logarithms. You know, actually indices, surds and logarithms are closely related.They are most of the time,studied together. So now, lets begin…

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Mathematics is the way

2

GRADE X

1.1

INDICES

A. PROPERTIES OF EXPONENT Have you ever bought eggs ? How many eggs that you get, if you buy 8 eggs each day for 8 days?

Recurring Number Magic Ok. This is easy question, isn’t it? So, what is the result? Yups, that is right 64. Do the result 64 comes from 8 x 8 ?. We can write 8 x 8 = 82. It means that we have used indices. What is indices ?

Activity: You write down the following 8 digit number on a piece of paper: 12345679

Now, Lets discuss it.

Then ask a friend to circle one of the digits. Say that they circle number 7.

Indices or powers are also called exponent The exponent of a number says how many times to use the number in multiplication.

You then ask your friend to multiply the 8 digit number by 63, and magically the result ends up being:

In example 82 = 8 x 8 = 64. How to read 82? Don’t miss it !

12345679 x 63 777777777

In words: 82 could be called ―8 to the second power‖, ―8 to the power 2‖ or simply ―8 squared‖.

with the answer as a row of the chosen number 7. How about if your friends circle number 3? Ask them to multiply by 27 and the result is 333333333. What is the secret ?

Need more example? Here they are. 35 = 3 x 3 x 3 x 3 x 3 In words : 35could be called “ 3 to the fifth power”, “3 to the power 5”. And exponents make it easier to write and use many multiplications. Example :

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GRADE X 116 is easier to write and read than 11 x 11 x 11x 11 x 11x 11 You can multiply any number by itself as many times as you want using this notation. So, in general:

Puzzle THE KEY OF EXPONENT The "Laws of Exponents" (also called "Rules of Exponents"), all come from three following ideas: 1. The exponent of a number says to multiply the number by itself so many times 2. The opposite of multiplying is dividing, so a negative exponent means divide

9 + 5 = 14.

3. A fractional exponent like 1/n means to take the nth root:

If you understand those, trust me that you are able to continue next journey of exponent. All the laws below are based on those ideas.

Laws of Exponents Here are the Laws (explanations follow): Law x = x x0 = 1 x-1 = 1/x

Example 61 = 6 70 = 1 4-1 = ¼

xmxn = xm+n xm/xn = xm-n (xm)n = xmn (xy)n = xnyn (x/y)n = xn/yn x-n = 1/xn

x2x3 = x2+3 = x5 x4/x2 = x4-2 = x2 (x2)3 = x2×3 = x6 (xy)3 = x3y3 (x/y)2 = x2 / y2 x-3 = 1/x3

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Mathematics is the way

Math Aptitude Test Which of the following sentences is correct? Nine and five are thirteen. or Nine and five is thirteen. Solution: Neither is correct ,

4

GRADE X

Laws Explained The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents.

Have a look at this example: Simplify and write down in positive exponents 1.

The example use “y and a” as variables not x. It can be changed to another alphabet

y 5 . y 7 y 2 1  2  y  2 2  y  4  4 2 y y y



3



 

 

2 a 2 2 3 2. 2a 3 .     2  a 3 . a  2 1 2  2 2 a  6 . a 3 2 3

3

 2 2  ( 3). a 6 (3)  2 1 a 3 

1 2a 3

3. Evaluate the following without using calculator : 4

(a)

83 3 8 2 3

 27     3  125   125 

b 27 

2

 3  

The question can be solved using the formula

2

3 2   9  3     3  3   3            5     5    5  25   3

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Did You Know

If you find it hard to remember all of these rules, then remember this: “ You can always work them out if you understand the three that have explained”.

The Strange case of 00

Ups, what will happen if x (variable) = 0 Positive Exponent (n>0) 0n = 0 Negative Exponent (n