Industrial Mechatronics Technology professionals in Kenya require strong algebraic skills to analyze and solve problems related to automation systems, control circuits, and mechanical-electrical integrations. Mastery of indices is essential for simplifying expressions involving powers, which frequently arise in calculations of sensor outputs, signal processing, and machine parameter estimations. This chapter focuses on applying algebraic rules of indices to manipulate and evaluate expressions efficiently in an industrial mechatronics context.

1.1 Indices

Indices represent the power to which a number or variable is raised. Understanding indices allows engineers to simplify calculations involving repeated multiplication, such as torque calculations, electrical signal scaling, and system dynamics modeling. The following sections explore the zero power, negative powers, fractional powers, and fundamental laws of indices with worked examples relevant to industrial mechatronics applications.

1.1.1 Power zero

Any non-zero number raised to the power zero equals one. This principle is critical when simplifying expressions in control system equations or sensor calibration formulas where terms may cancel out. The rule is expressed as:

$$a^0 = 1 \quad \text{for} \quad a \neq 0$$

Worked Examples

Example 1: A sensor output voltage is modeled as \(V = 5 \times 10^0\) volts. Calculate \(V\).

Given: \(a = 10\), exponent \(0\)

$$a^0 = 1$$

$$V = 5 \times 1$$

Answer: 5 \text{ volts}

Example 2: In a robotic arm control system, the gain factor is \(G = 3^0\). Find \(G\).

Given: \(a = 3\), exponent \(0\)

$$3^0 = 1$$

Answer: 1

Example 3: Calculate the value of \((12)^0\) used in a PLC logic timing formula.

Given: \(a = 12\), exponent \(0\)

$$12^0 = 1$$

Answer: 1

Example 4: For a signal processing algorithm, the term \( (x)^0 \) appears where \(x = 7.5\). Find the value.

Given: \(a = 7.5\), exponent \(0\)

$$7.5^0 = 1$$

Answer: 1

Example 5: During calibration, the correction factor is expressed as \( (0.25)^0 \). Compute the correction factor.

Given: \(a = 0.25\), exponent \(0\)

$$0.25^0 = 1$$

Answer: 1

1.1.2 Negative powers

Negative powers represent the reciprocal of the base raised to the corresponding positive power. This concept is frequently used in industrial mechatronics when calculating inverse relationships, such as resistance in parallel circuits or decay rates in sensor signals. The rule is:

$$a^{-n} = \frac{1}{a^n} \quad \text{for} \quad a \neq 0$$

Worked Examples

Example 1: Calculate \(5^{-2}\) for a voltage attenuation factor.

Given: \(a = 5\), \(n = 2\)

$$5^{-2} = \frac{1}{5^2}$$

$$= \frac{1}{25}$$

Answer: 0.04

Example 2: In a motor speed controller, the factor \( (2)^{-3} \) appears. Find its value.

Given: \(a = 2\), \(n = 3\)

$$2^{-3} = \frac{1}{2^3}$$

$$= \frac{1}{8}$$

Answer: 0.125

Example 3: For a sensor decay process, calculate \( (10)^{-1} \).

Given: \(a = 10\), \(n = 1\)

$$10^{-1} = \frac{1}{10}$$

Answer: 0.1

Example 4: Determine the value of \( (0.5)^{-2} \) used in signal amplification.

Given: \(a = 0.5\), \(n = 2\)

$$0.5^{-2} = \frac{1}{(0.5)^2}$$

$$= \frac{1}{0.25}$$

Answer: 4

Example 5: Calculate \( (3)^{-4} \) for a component gain factor.

Given: \(a = 3\), \(n = 4\)

$$3^{-4} = \frac{1}{3^4}$$

$$= \frac{1}{81}$$

Answer: \approx 0.0123

1.1.3 Fractional powers

Fractional powers represent roots of numbers or variables. This is important in mechatronics for calculating parameters like RMS values, natural frequencies, and material properties. The general rule:

$$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

Worked Examples

Example 1: Calculate \(27^{\frac{1}{3}}\) for a gear ratio computation.

Given: \(a = 27\), \(m=1\), \(n=3\)

$$27^{\frac{1}{3}} = \sqrt[3]{27}$$

$$= 3$$

Answer: 3

Example 2: Find \(16^{\frac{3}{4}}\) used in a signal power calculation.

Given: \(a=16\), \(m=3\), \(n=4\)

$$16^{\frac{3}{4}} = (\sqrt[4]{16})^3$$

$$= (2)^3$$

$$= 8$$

Answer: 8

Example 3: Calculate \(81^{\frac{1}{2}}\) for a motor torque formula.

Given: \(a=81\), \(m=1\), \(n=2\)

$$81^{\frac{1}{2}} = \sqrt{81}$$

$$= 9$$

Answer: 9

Example 4: Determine \(125^{\frac{2}{3}}\) for a sensor calibration curve.

Given: \(a=125\), \(m=2\), \(n=3\)

$$125^{\frac{2}{3}} = (\sqrt[3]{125})^2$$

$$= (5)^2$$

$$= 25$$

Answer: 25

Example 5: Find \(9^{\frac{3}{2}}\) used in mechanical stress calculations.

Given: \(a=9\), \(m=3\), \(n=2\)

$$9^{\frac{3}{2}} = (\sqrt{9})^3$$

$$= (3)^3$$

$$= 27$$

Answer: 27

1.1.4 Laws of indices

The laws of indices allow for the simplification of expressions involving powers, combining or separating terms efficiently in control algorithms, mechanical system modeling, and electrical circuit analysis. The main laws are:

1. Product Law: \(a^m \times a^n = a^{m+n}\)
2. Quotient Law: \(\frac{a^m}{a^n} = a^{m-n}\), \(a \neq 0\)
3. Power of a Power: \((a^m)^n = a^{mn}\)
4. Power of a Product: \((ab)^n = a^n b^n\)
5. Power of a Quotient: \((\frac{a}{b})^n = \frac{a^n}{b^n}\), \(b \neq 0\)

Worked Examples

Example 1: Simplify \(2^3 \times 2^4\) for a signal gain calculation.

Given: \(a=2\), \(m=3\), \(n=4\)

$$2^3 \times 2^4 = 2^{3+4}$$

$$= 2^7$$

$$= 128$$

Answer: 128

Example 2: Simplify \(\frac{5^6}{5^2}\) for a motor power ratio.

Given: \(a=5\), \(m=6\), \(n=2\)

$$\frac{5^6}{5^2} = 5^{6-2}$$

$$= 5^4$$

$$= 625$$

Answer: 625

Example 3: Calculate \((3^2)^4\) in a control system's exponential decay model.

Given: \(a=3\), \(m=2\), \(n=4\)

$$(3^2)^4 = 3^{2 \times 4}$$

$$= 3^8$$

$$= 6561$$

Answer: 6561

Example 4: Simplify \((4 \times 5)^3\) for a combined actuator force calculation.

Given: \(a=4\), \(b=5\), \(n=3\)

$$(4 \times 5)^3 = 4^3 \times 5^3$$

$$= 64 \times 125$$

$$= 8000$$

Answer: 8000

Example 5: Simplify \((\frac{6}{2})^4\) for a gear speed ratio.

Given: \(a=6\), \(b=2\), \(n=4\)

$$(\frac{6}{2})^4 = \frac{6^4}{2^4}$$

$$= \frac{1296}{16}$$

$$= 81$$

Answer: 81

Chapter Summary

This chapter covered the application of algebra in engineering mathematics, beginning with the concept of indices including zero, negative, and fractional powers along with the fundamental laws of indices such as addition, subtraction, division, and multiplication. The importance of BODMAS was emphasized for correctly evaluating expressions involving multiple operations. The chapter then explored roots, focusing on square, cube, and nth roots and their calculations. Logarithms were introduced with their key laws: the product, quotient, and power laws, enabling simplification of complex expressions. The use of scientific calculators was explained, highlighting functions such as power on/off, mode settings including degree, radian, and gradient, as well as clear, save, and shift operations. Methods for solving simultaneous equations with up to three unknowns were discussed through elimination, substitution, reduction, and graphical techniques. Finally, quadratic equations were addressed, covering factorization, the quadratic formula, completing the square, and graphical solutions, providing a comprehensive toolkit for algebraic problem solving in engineering contexts.