Theory of Indices: Laws, Rules & Formulae

1. Meaning of Base and Index

If a is any real number and n is a positive integer, the repeated product:

a × a × a × … (up to n times) = aⁿ

Here:

• a is called the Base
• n is called the Index, Exponent, or Power

2. Fundamental Laws of Indices

The syllabus requires preliminary knowledge of index formulae. These six foundational algebraic laws govern all exponential expressions:

3. Step-by-Step Practice Simplifications

Problem 1: Simplifying an Index Expression

Question: Simplify the algebraic fraction:

E = [ 2^(n+4) - (2 × 2^n) ] ÷ [ 2 × 2^(n+3) ]

Step-by-Step Solution:

1. Expand using the product rule [ 2^(n+k) = 2^n × 2^k ]:
   • In the numerator: 2^(n+4) = 2^n × 2^4 = 16 × 2^n
   • Second term: 2 × 2^n = 2^1 × 2^n = 2 × 2^n
   • Numerator becomes: (16 × 2^n) - (2 × 2^n) = (16 - 2) × 2^n = 14 × 2^n
2. Expand the denominator:
   • 2 × 2^(n+3) = 2^1 × 2^n × 2^3 = 2^n × (2 × 8) = 16 × 2^n
3. Cancel the common term 2^n from both numerator and denominator:
   • E = (14 × 2^n) ÷ (16 × 2^n)
   • E = 14 ÷ 16
4. Reduce the fraction:
   • Divide numerator and denominator by 2:
   • E = 7 / 8 (or 0.875)

Problem 2: Solving an Exponential Equation for x

Question: Solve for x:

9^(x+2) = 240 + 9^x

Step-by-Step Solution:

1. Rewrite 9^(x+2) as 9^x × 9²:
   • Since 9² = 81, 9^(x+2) = 81 × 9^x
2. Substitute into the equation:
   • 81 × 9^x = 240 + 9^x
3. Transpose 9^x to the left-hand side:
   • (81 × 9^x) - (1 × 9^x) = 240
   • 80 × 9^x = 240
4. Divide both sides by 80:
   • 9^x = 240 ÷ 80 = 3
5. Convert both sides to a common base (Base 3):
   • Since 9 = 3², we have (3²)^x = 3¹
   • 3^(2x) = 3¹
6. Equate the powers (since the bases are identical):
   • 2x = 1
   • x = 1 / 2 (or 0.5)