Mechatronics Technology professionals in Kenya regularly engage with complex systems requiring precise mathematical tools to analyse and optimise performance. Understanding and manipulating indices is foundational for handling exponential data in sensor outputs, control algorithms, and signal processing. This chapter develops your ability to apply indices effectively in engineering calculations, preparing you for advanced statistical and probabilistic analyses in mechatronics applications.

1.1 Indices

Indices, also known as exponents or powers, describe repeated multiplication of a base number. Mastery of indices is essential in mechatronics technology for simplifying equations involving growth rates, decay processes, and scaling laws within sensors and actuators. This section covers fundamental index rules including zero, negative, and fractional powers, culminating in the laws that govern their manipulation.

1.1.1 Power zero

The zero power rule states that any non-zero number raised to the power of zero equals one. This property is crucial when simplifying expressions in control systems and signal processing where terms may cancel out or normalize.

The general formula is:

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

Worked Examples

Example 1: Calculate \(5^0\).

Given: \(a = 5\)

$$a^0 = 1$$

$$5^0 = 1$$

Answer: 1

Example 2: Evaluate \((12)^0\).

Given: \(a = 12\)

$$a^0 = 1$$

$$12^0 = 1$$

Answer: 1

Example 3: Find \((0.8)^0\).

Given: \(a = 0.8\)

$$a^0 = 1$$

$$0.8^0 = 1$$

Answer: 1

Example 4: Determine \((-7)^0\).

Given: \(a = -7\)

$$a^0 = 1$$

$$(-7)^0 = 1$$

Answer: 1

Example 5: Calculate \((\frac{3}{4})^0\).

Given: \(a = \frac{3}{4}\)

$$a^0 = 1$$

$$(\frac{3}{4})^0 = 1$$

Answer: 1

1.1.2 Negative powers

Negative powers indicate the reciprocal of the base raised to the corresponding positive power. This concept is pivotal in mechatronics for inverse relationships such as impedance or gain calculations in circuits.

The rule is:

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

Worked Examples

Example 1: Calculate \(4^{-2}\).

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

$$a^{-n} = \frac{1}{a^n}$$

$$4^{-2} = \frac{1}{4^2}$$

$$= \frac{1}{16}$$

Answer: \(\frac{1}{16}\)

Example 2: Evaluate \((2)^{-3}\).

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

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

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

Answer: \(\frac{1}{8}\)

Example 3: Find \((\frac{1}{5})^{-2}\).

Given: \(a=\frac{1}{5}, n=2\)

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

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

$$= 25$$

Answer: 25

Example 4: Determine \(10^{-1}\).

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

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

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

Answer: \(\frac{1}{10}\)

Example 5: Calculate \((-3)^{-2}\).

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

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

$$= \frac{1}{9}$$

Answer: \(\frac{1}{9}\)

1.1.3 Fractional powers

Fractional powers represent roots of numbers, where the denominator of the fraction is the root index. This operation is fundamental in mechatronics when dealing with calibration curves, sensor response linearization, and signal transformations.

The rule is:

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

Worked Examples

Example 1: Calculate \(9^{\frac{1}{2}}\).

Given: \(a=9, m=1, n=2\)

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

$$9^{\frac{1}{2}} = \sqrt{9}$$

$$= 3$$

Answer: 3

Example 2: Evaluate \(27^{\frac{1}{3}}\).

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

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

$$= 3$$

Answer: 3

Example 3: Find \(16^{\frac{3}{4}}\).

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

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

$$\sqrt[4]{16} = 2$$

$$2^3 = 8$$

Answer: 8

Example 4: Determine \(81^{\frac{1}{2}}\).

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

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

$$= 9$$

Answer: 9

Example 5: Calculate \(125^{\frac{2}{3}}\).

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

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

$$\sqrt[3]{125} = 5$$

$$5^2 = 25$$

Answer: 25

1.1.4 Laws of indices

The laws of indices allow the simplification and manipulation of expressions with the same base, which is essential in mechatronics for solving exponential equations in control systems, signal processing, and electronics.

Key laws:

1. Product law: \(a^m \times a^n = a^{m+n}\)
2. Quotient law: \(\frac{a^m}{a^n} = a^{m-n}\)
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: \((\frac{a}{b})^n = \frac{a^n}{b^n}\)

Worked Examples

Example 1: Simplify \(3^2 \times 3^4\).

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

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

$$= 3^6$$

$$= 729$$

Answer: 729

Example 2: Simplify \(\frac{5^7}{5^3}\).

Given: \(a=5, m=7, n=3\)

$$\frac{5^7}{5^3} = 5^{7-3}$$

$$= 5^4$$

$$= 625$$

Answer: 625

Example 3: Simplify \((2^3)^4\).

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

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

$$= 2^{12}$$

$$= 4096$$

Answer: 4096

Example 4: Simplify \((3 \times 4)^2\).

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

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

$$= 9 \times 16$$

$$= 144$$

Answer: 144

Example 5: Simplify \((\frac{6}{2})^3\).

Given: \(a=6, b=2, n=3\)

$$(\frac{6}{2})^3 = \frac{6^3}{2^3}$$

$$= \frac{216}{8}$$

$$= 27$$

Answer: 27

Chapter Summary

This chapter covered fundamental concepts in Engineering Mathematics II, beginning with indices, including zero, negative, and fractional powers, and the essential laws governing their addition, subtraction, division, and multiplication. The order of operations was emphasized through BODMAS to ensure correct evaluation of expressions. Roots were explored next, focusing on square roots, cube roots, and general nth roots. Logarithms were introduced with their key laws: product, quotient, and power rules, which simplify complex calculations. The use of scientific calculators was detailed, highlighting functions such as power on/off, mode settings for degree, radian, gradient, and statistical data, as well as clearing, saving, and shift functions. Techniques for solving simultaneous equations with up to three variables were thoroughly explained, including elimination, substitution, reduction, and graphical methods. Finally, quadratic equations were addressed through factorization, the quadratic formula, completing the square, and graphical solutions, providing a comprehensive toolkit for solving a variety of algebraic problems in engineering contexts.