Algebra is a foundational tool for industrial control and installation professionals, enabling precise calculation of component values, system parameters, and performance metrics. In Kenya’s industrial sector, algebraic skills are essential for designing control systems, troubleshooting electrical circuits, and optimizing installations. Mastery of bases and indices underpins calculations involving exponential growth, decay, and power ratings, all critical for safe and efficient industrial operations.

1.1 Base and Index

Algebraic expressions involving bases and indices are common in engineering calculations, such as determining electrical loads, sizing equipment, and calculating rates of change. The laws of indices simplify complex computations and are vital for interpreting technical specifications and engineering drawings.

1.1.1 Meaning of Base and Index

A base is the number being multiplied, while an index (or exponent) indicates how many times the base is used as a factor. In industrial control, these are used to express power ratings, scaling factors, and system amplification.

The general form is:$$ a^{n} $$ where \( a \) is the base and \( n \) is the index.

Worked Examples

Example 1:
A control panel requires calculating the power of a signal amplifier. If the gain is \( 2^3 \), what is the amplified signal strength?

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

$$ a^{n} = 2^{3} $$$$ 2^{3} = 2 \times 2 \times 2 $$$$ 2 \times 2 = 4 $$$$ 4 \times 2 = 8 $$ Answer: 8 units

Example 2:
An industrial installation uses a motor rated at \( 5^2 \) watts. What is the power output?

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

$$ a^{n} = 5^{2} $$$$ 5^{2} = 5 \times 5 $$$$ 5 \times 5 = 25 $$ Answer: 25 watts

Example 3:
A PLC (Programmable Logic Controller) memory module calculates \( 10^4 \) memory addresses. How many unique addresses are available?

Given: \( a = 10 \), \( n = 4 \)

$$ a^{n} = 10^{4} $$$$ 10^{4} = 10 \times 10 \times 10 \times 10 $$$$ 10 \times 10 = 100 $$$$ 100 \times 10 = 1000 $$$$ 1000 \times 10 = 10000 $$ Answer: 10,000 addresses

Example 4:
A sensor signal is multiplied by \( 3^4 \) in a calibration process. What is the resulting amplification?

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

$$ a^{n} = 3^{4} $$$$ 3 \times 3 = 9 $$$$ 9 \times 3 = 27 $$$$ 27 \times 3 = 81 $$ Answer: 81 times amplification

Example 5:
A technician needs to compute the total number of possible combinations for a 2-position switch with 6 switches: \( 2^6 \).

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

$$ a^{n} = 2^{6} $$$$ 2 \times 2 = 4 $$$$ 4 \times 2 = 8 $$$$ 8 \times 2 = 16 $$$$ 16 \times 2 = 32 $$$$ 32 \times 2 = 64 $$ Answer: 64 combinations

1.1.2 Laws of Indices

The laws of indices help simplify calculations involving powers and are widely used in engineering mathematics.

• Multiplication: \( a^{m} \times a^{n} = a^{m+n} \)
• Division: \( \frac{a^{m}}{a^{n}} = a^{m-n} \)
• Power of a Power: \( (a^{m})^{n} = a^{m \times n} \)
• Fractional Index: \( a^{\frac{m}{n}} = \sqrt[n]{a^{m}} \)
• Negative Index: \( a^{-n} = \frac{1}{a^{n}} \)
• Zero Index: \( a^{0} = 1 \)

Worked Examples

Example 1:
A relay circuit uses \( 4^{3} \times 4^{2} \) resistors. Find the total equivalent resistance.

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

$$ a^{m} \times a^{n} = a^{m+n} = 4^{3+2} $$$$ 4^{5} = 4 \times 4 \times 4 \times 4 \times 4 $$$$ 4 \times 4 = 16 $$$$ 16 \times 4 = 64 $$$$ 64 \times 4 = 256 $$$$ 256 \times 4 = 1024 $$ Answer: 1024 units

Example 2:
A control system divides signal voltage: \( \frac{7^{5}}{7^{2}} \). Calculate the output voltage.

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

$$ \frac{a^{m}}{a^{n}} = a^{m-n} = 7^{5-2} $$$$ 7^{3} = 7 \times 7 \times 7 $$$$ 7 \times 7 = 49 $$$$ 49 \times 7 = 343 $$ Answer: 343 volts

Example 3:
A PLC stores data: \( (2^{4})^{3} \). Find the total storage locations.

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

$$ (a^{m})^{n} = a^{m \times n} = 2^{4 \times 3} $$$$ 2^{12} = 4096 $$ Answer: 4096 locations

Example 4:
A technician calculates the root of a resistance value: \( 9^{\frac{1}{2}} \).

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

$$ a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = \sqrt{9} = 3 $$ Answer: 3 ohms

Example 5:
A safety device is rated at \( 5^{-2} \) amps. Find the reciprocal value.

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

$$ a^{-n} = \frac{1}{a^{n}} = \frac{1}{5^{2}} $$$$ 5^{2} = 25 $$$$ \frac{1}{25} = 0.04 $$ Answer: 0.04 amps

1.1.3 Application of Indices in Industrial Control Calculations

Indices are applied in calculations such as determining equipment power, exponential decay in signal processing, and scaling operations in control systems.

Worked Examples

Example 1:
A frequency converter increases output frequency exponentially: \( f_{out} = f_{in} \times 2^{n} \), where \( f_{in} = 50 \) Hz, \( n = 2 \).

Given: \( f_{in} = 50 \) Hz, \( n = 2 \)

$$ f_{out} = 50 \times 2^{2} $$$$ 2^{2} = 4 $$$$ 50 \times 4 = 200 $$ Answer: 200 Hz

Example 2:
A sensor signal decays exponentially: \( S = 100 \times 0.5^{3} \). Find the signal strength after 3 cycles.

Given: \( S_0 = 100 \), \( a = 0.5 \), \( n = 3 \)

$$ S = 100 \times 0.5^{3} $$$$ 0.5 \times 0.5 = 0.25 $$$$ 0.25 \times 0.5 = 0.125 $$$$ 100 \times 0.125 = 12.5 $$ Answer: 12.5 units

Example 3:
A county water treatment plant uses pumps with power scaling: \( P = 3^{4} \) kW. Calculate total power.

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

$$ P = 3^{4} = 3 \times 3 \times 3 \times 3 $$$$ 3 \times 3 = 9 $$$$ 9 \times 3 = 27 $$$$ 27 \times 3 = 81 $$ Answer: 81 kW

Example 4:
A digital security system uses \( 2^{8} \) possible codes. How many codes are available?

Given: \( a = 2 \), \( n = 8 \)

$$ 2^{8} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 $$$$ 2 \times 2 = 4 $$$$ 4 \times 2 = 8 $$$$ 8 \times 2 = 16 $$$$ 16 \times 2 = 32 $$$$ 32 \times 2 = 64 $$$$ 64 \times 2 = 128 $$$$ 128 \times 2 = 256 $$ Answer: 256 codes

Example 5:
A university lab uses exponential growth to model bacteria: \( N = 5 \times 3^{2} \). Find the population after 2 hours.

Given: \( N_0 = 5 \), \( a = 3 \), \( n = 2 \)

$$ N = 5 \times 3^{2} $$$$ 3 \times 3 = 9 $$$$ 5 \times 9 = 45 $$ Answer: 45 bacteria

Chapter Summary

This chapter began by reviewing the concepts of base and index, forming the foundation for understanding powers and exponents in algebraic expressions. It then explored the laws of indices, providing systematic rules for manipulating expressions with exponents, and applied these principles to solve indicial equations commonly encountered in engineering contexts. The discussion advanced to the laws of logarithms, equipping students with strategies to simplify and manipulate logarithmic expressions, followed by step-by-step solutions to logarithmic equations. Techniques for converting between different bases were introduced, enhancing flexibility in handling both indices and logarithms. Practical skills in using scientific calculators for evaluating powers, roots, and logarithmic values were also covered. The chapter continued with methods for reducing complex equations to simpler forms, including the reduction and solution of equations that can be expressed in quadratic form. It concluded with systematic approaches for solving simultaneous linear equations in three unknowns and applied these algebraic skills to real-world problems involving arithmetic and geometric progressions, reinforcing the practical value of algebra in engineering mathematics.