Lesson Title: Laws of Indices.

Objectives :

At the end of this lesson, you will be able to,

1. Identify the laws of indices - Multiplication, Division, and Product law.

2. Solve problems relating to laws of indices.

3. Understand the concepts behind Zero Indices, Negative Indices, and Fractional Indices.

What you should know :

You should be familiar with BODMAS precedence and other basic arithmetic operations.

Introduction :

The word indices is the plural form of an index. Index a is the number of times a number n must multiply itself to equal a certain number b, that is,

n^a = b.

Presentation /Steps

1. The Multiplication Law.

=> aⁿ * aⁿ * aⁿ = aⁿ⁺ⁿ⁺ⁿ Where the base values a are the same.

=> a³ * a² = a³⁺² = a⁵

=> 10² * 10 = 10²⁺¹ = 10³

N.B any value raised to the index of 1 is still that value.

=> 2a² * 3a⁴ * 4a² = (2*3*4)a²⁺⁴⁺²

^N.B 2a is the same as 2*a likewise 3*a, and 4*a, since a is the affected value and they are similar, then a²⁺⁴⁺² ,

Therefore, the final answer will be,

(24)a⁸ = 24a⁸ .

2. The Division Law.

=> In this law, the indices are subtracted, unlike multiplication law where the indices are added.

=> aⁿ ÷ aⁿ = a^n-n . Where the base values a are the same.

=> C⁴ ÷ C² = C^4-2 = C²

=> 3⁷ ÷ 3² = 3^7-2 = 3⁵

=> 24b⁹ ÷ 4b³ = (24 ÷ 4)b^9-3 = 6b⁶

3. The Product Laws.

=> In this law, the index outside the bracket multiplies the indices inside the bracket.

=> N:B, A value raised to an index 1 is still the same value, so there is no need to write the index '1'.

=> (X^a)^b = X^a*b

=> (2x^a)^b is the same as 2^b * X^a*b = 2^bx^a*b

=> (g^-3)² = g^-3*2 = g^-6

=> (-4b)² = (-4)² * b² ==> -4 * -4 = 8 * b² = 8b²

=> (-4b)³ = (-4)³ * b^{3 ==>}

-4 * -4 * -4 = -32 * b³ = -32b³

N:B If a negative number is raised to an odd index, the number remains negative,

but, if a negative number is raised to an even index, the number changes to a positive number.

4. Zero Indices.

A value raised to the index of 0 equals 1.

a⁰ = 1.

=> a^-2 * a² = a^{-2 + 2} = a⁰ = 1.

=> b³ ÷ b³ = b^{3 - 3} = b⁰ = 1.

5. Negative Indices.

=> a^-x = ¹/a^x

=> 2a^-x = (2)¹/a^x = ²/a^x

=> The negative index is often used when the final result of an expression has a negative index.

6. Fractional Indices.

=> X^1/a , X^c/b

=> In this case, the fractional part is used for the calculation, the denominator of the expressions above (a, b) is used as the root of the value x while (1, c) is used as the power index.

=> That is (a√x)¹, (b√x)^c

=> 4½ = √4 = ± 2.

N:B "2" is not indicated as the root value because √ represents the square root by default, you can only include a root value when the value is above 2.

=> 27⅔ = (3√27)² = (3)² = ± 9.

Summary / Conclusion:

In this lesson, we were able to solve problems by applying the laws of indices and other indices sub-laws.

=> Indices help in writing LCMs of a number in a short form: the LCM of 16 = 2 * 2 * 2 * 2 which can be shortened as 2⁴

Exercise :

Express the following 8^-⅔ , (a³b)⁴