01Aindexlaws

= Introduction to Index Laws =

Literacy and Notation

 * Index Form **
 * For example, 3 4.
 * The** __base__ **is the main number (eg 3)
 * The** __index__ **is the small superscript number (eg 4)
 * The** __index__ **is also called the** __power__ **or the** __exponent__ **
 * The index tells us how many times to multiply the base by itself.


 * Factor Form **
 * For example, 3 × 3 × 3 × 3
 * The base has been multiplied by itself the amount of times given by the index


 * Basic Numeral **
 * For example, 81
 * The result of evaluating the multiplication.

... ** a) ** . Write 5 2 a 3 in factor form
 * Example 1 **

... ... ... ... 5 2 a 3 = 5 × 5 × a × a × a

... ** b) ** . Write 7 × a × a × a × a × a × b × b in index form.

... ... ...... 7 × a × a × a × a × a × b × b = 7a 5 b 2.

... ** c) ** . Write 8 in factor form __and__ index form using 2 as the base.

... ... ... ... 8 = 2 × 2 × 2 = 2 3.

Factor Trees
To write a larger number in index form (or as a product of its prime factors), draw a factor tree.
 * Divide each number into a pair of factors
 * The ends of each branch will be the factors of the original number


 * Example 2 **

... ... Write 360 as a product of its prime factors and in index form.



... ... Hence 360 = 2 × 2 × 2 × 3 × 3 × 5

... ... Hence 360 = 2 3 × 3 2 × 5

Evaluating Numbers in Index Form
Recall ** BOMDAS **
 * ** B ** = Brackets
 * ** O ** = Of ... {as in "Power Of a number"}
 * ** MD ** = Multiplication and Division
 * ** AS ** = Addition and Subtraction

The "Power Of" is more important than Mulitiply and Divide,
 * so evaluate any powers __first__
 * __then__ multiply or divide the results


 * Example 3 **

... ... Evaluate 4a 3 when a = 5

... ... ... 4a 3 = 4 × 5 3 ... ... ... ... .. = 4 × 125 ... ... ... ... .. = 500

Multiplying Numbers in Index Form



 * 1st Index Law **

... When __**multiplying**__ numbers in index form **__with the same base__**:
 * keep the base and __**ADD**__ the indices


 * Example 4 **

... Simplify each of the following: ... ** 1) ** . a 5 × a 4 ... ... .... = a 9

... ** 2) ** . 5b 6 × 2b 2 ... ... ... = 10b 8

... ** 3) ** . a 4 b 8 × a 4 b ... ... ... = a 8 b 9


 * Notes: **
 * In ** (2) **the 5 and 2 are normal numbers (not in index form)
 * so multiply as normal : 5 × 2 = 10
 * In ** (3) ** the rule says "with the same base" so combine the a's and then combine the b's
 * In ** (3) **the second b has an invisible index of 1
 * so add 8 + 1 = 9


 * Dividing Numbers in Index Form **



... When __**dividing**__ numbers in index form **__with the same base__**:
 * 2nd Index Law **
 * keep the base and __**SUBTRACT**__ the indices


 * Example 5 **

... Simplify each of the following:

math . \qquad \textbf{1)} \quad a^7 \div a^3 \\ . \\ . \qquad \qquad = \dfrac{a^7}{a^3} \\ . \\ . \qquad \qquad = a^4 math . . math . \qquad \textbf{2)} \quad 10a^5 \div 2a^4 \\. \\ . \qquad \qquad = \dfrac{10a^5}{2a^4} \\. \\ . \qquad \qquad = 5a^1 \\. \\ . \qquad \qquad = 5a math . . math . \qquad \textbf{3)} \quad a^9b^6 \div a^3b \\ . \\ . \qquad \qquad =\dfrac{a^9b^6}{a^3b} \\ . \\ . \qquad \qquad = a^6b^5 math . . math . \qquad \textbf{4)} \quad 8a^7b^3 \div 6a^2b^8 \\. \\ . \qquad \qquad = \dfrac{8a^7b^3}{6a^2b^8} \\. \\ . \qquad \qquad = \dfrac{4a^5}{3b^5} math


 * Notes: **
 * In ** (2) **the 10 and 2 are normal numbers (not in index form)
 * so divide as normal : 10 ÷ 2 = 5
 * In ** (3) ** the rule says "with the same base" so combine the a's and then combine the b's
 * In ** (3) **the second b has an invisible index of 1
 * so 6 – 1 = 5
 * In ** (4) ** treat the 8 over 6 as a normal fraction, so simplify it to get 4 over 3
 * In ** (4) ** the a's on the bottom cancel, leaving a 5 on the top of the fraction
 * In ** (4) ** the b's on the top cancel, leaving b 5 on the bottom of the fraction

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