Recap of Basic Index Laws & Zero Index
Unit 2 • Lesson 01 of 05
Recap of Basic Index Laws & Zero Index
Grade 11 Mathematics • G.C.E. O/L Sri Lanka Local Syllabus
Learning Objectives
- Recall and apply the five fundamental laws of indices introduced in Grade 10.
- Understand and apply the zero index rule (\(a^0 = 1\), where \(a \neq 0\)).
- Simplify multi-variable algebraic monomials accurately.
Core Laws of Indices
| Law Name | Algebraic Rule | Numerical Example |
|---|---|---|
| Product Law | \(a^m \times a^n = a^{m+n}\) | \(2^3 \times 2^4 = 2^{3+4} = 2^7\) |
| Quotient Law | \(a^m \div a^n = a^{m-n} \quad (a \neq 0)\) | \(5^6 \div 5^2 = 5^{6-2} = 5^4\) |
| Power of a Power | \((a^m)^n = a^{mn}\) | \((3^2)^4 = 3^{2 \times 4} = 3^8\) |
| Power of a Product | \((ab)^n = a^n b^n\) | \((2x)^3 = 2^3 x^3 = 8x^3\) |
| Power of a Quotient | \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)\) | \(\left(\frac{x}{3}\right)^2 = \frac{x^2}{3^2} = \frac{x^2}{9}\) |
| Zero Index Law | \(a^0 = 1 \quad (a \neq 0)\) | \(7^0 = 1, \quad (4x^2)^0 = 1\) |
Step-by-Step Worked Examples
Example 1: Simplify \(\frac{(2x^3)^2 \times x^0}{4x^4}\)
- Apply the power of a product rule to \((2x^3)^2\) and the zero index rule to \(x^0\): \[\frac{2^2 \cdot (x^3)^2 \times 1}{4x^4} = \frac{4x^6}{4x^4}\]
- Cancel the common numerical factor \(4\) and apply the quotient law: \[x^{6-4} = x^2\]
Example 2: Evaluate \(3x^0 + (3x)^0 - 2^0\) (where \(x \neq 0\))
- Note that in \(3x^0\), only \(x\) has exponent \(0\), whereas in \((3x)^0\), the entire bracket has exponent \(0\): \[3(1) + 1 - 1\]
- Calculate the final value: \[3 + 1 - 1 = 3\]
Practice Drill
1. Simplify: \((3a^2b)^3 \div (9a^4b^2)\)
Show Step-by-Step Answer
Step 1: Expand the first bracket using the power of a product rule:
\((3a^2b)^3 = 3^3 \cdot (a^2)^3 \cdot b^3 = 27a^6b^3\)
Step 2: Divide by \(9a^4b^2\):
\(\frac{27a^6b^3}{9a^4b^2} = \frac{27}{9} \cdot a^{6-4} \cdot b^{3-2} = \mathbf{3a^2b}\)
2. Evaluate: \(5^0 + (2x)^0 - 3x^0\) (where \(x \neq 0\))
Show Step-by-Step Answer
Step 1: Replace each zero-power term with \(1\):
\(5^0 = 1\), \((2x)^0 = 1\), and \(3x^0 = 3 \times 1 = 3\)
Step 2: Combine the numbers:
\(1 + 1 - 3 = \mathbf{-1}\)
3. Simplify: \(\frac{4m^5n^3 \times 3m^2n^4}{6(mn)^3}\)
Show Step-by-Step Answer
Step 1: Multiply the numerator terms and expand the denominator:
\(\frac{12m^{5+2}n^{3+4}}{6m^3n^3} = \frac{12m^7n^7}{6m^3n^3}\)
Step 2: Apply the quotient law:
\(2m^{7-3}n^{7-3} = \mathbf{2m^4n^4}\)