By the end of this chapter, you will be able to: - Correctly perform calculations involving indices to meet task requirements. - Accurately carry out calculations using logarithms as needed for your work. - Use a scientific calculator confidently and correctly to solve mathematical problems. - Solve simultaneous equations accurately, following the steps required for each task. - Solve quadratic equations correctly to find the right solutions for given problems.
Mastering these algebra skills will help you tackle real-life technical challenges with confidence and precision in your trade.
Welding professionals frequently encounter calculations involving powers and exponents when determining quantities such as heat input, material thickness scaling, and welding current adjustments. Mastery of algebraic operations on indices is essential for accurate measurements and ensuring weld quality. This chapter focuses on applying algebraic rules related to indices, enabling welders to solve practical problems efficiently and precisely.
An index (plural indices, also called a power or an exponent) tells how many times a number is multiplied by itself. In 3⁴ the base is 3 and the index is 4, so 3⁴ = 3 × 3 × 3 × 3 = 81. Indices make long products short, and they appear in many welding and fabrication formulas, such as the area of a circle (π × r²), the volume of a cylinder and the heat input and strength calculations used in the workshop.
The rules of indices let a welder simplify an expression before calculating it, work with very large and very small quantities, and change between units such as mm² and m². The following sections cover the special cases of a zero index, a negative index and a fractional index, and then the laws that combine powers.
Any non-zero number raised to the power zero is equal to one. The rule follows from the law for dividing powers: aⁿ ÷ aⁿ = a^(n − n) = a⁰, and a number divided by itself is 1, so a⁰ = 1. The base must not be zero.
a⁰ = 1, where a ≠ 0
Example 1: Calculate 5⁰.
Given: a = 5
5⁰ = 1
Answer: 1
Example 2: Find the value of (12.7)⁰.
(12.7)⁰ = 1
Answer: 1
Example 3: Determine the value of (0.003)⁰.
(0.003)⁰ = 1
Answer: 1
Example 4: Simplify (7 ÷ 3)⁰.
(7 ÷ 3)⁰ = 1
Answer: 1
Example 5: Calculate x⁰ when x = 250.
x⁰ = 1, whatever the value of x (except 0)
Answer: 1
A negative index means the reciprocal of the same power with a positive index. It is a way of writing "one divided by" in short form, and it is useful when a quantity falls as another one rises, such as a current that falls as the resistance in a circuit rises.
a⁻ⁿ = 1 ÷ aⁿ, where a ≠ 0
A fraction raised to a negative power is turned upside down: (a ÷ b)⁻ⁿ = (b ÷ a)ⁿ.
Example 1: Calculate 2⁻³.
Given: a = 2, n = 3
2⁻³ = 1 ÷ 2³ = 1 ÷ 8
Answer: 1 ÷ 8 = 0.125
Example 2: Evaluate 10⁻².
10⁻² = 1 ÷ 10² = 1 ÷ 100
Answer: 1 ÷ 100 = 0.01
Example 3: Find the value of 5⁻¹.
5⁻¹ = 1 ÷ 5¹ = 1 ÷ 5
Answer: 1 ÷ 5 = 0.2
Example 4: Simplify (3 ÷ 4)⁻².
(3 ÷ 4)⁻² = (4 ÷ 3)² = 16 ÷ 9
Answer: 16 ÷ 9
Example 5: Calculate (0.5)⁻³.
(0.5)⁻³ = 1 ÷ (0.5)³ = 1 ÷ 0.125 = 8
Answer: 8
A fractional index is a root written as a power. The denominator of the fraction is the order of the root, and the numerator is the power to which the number is raised.
a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
So a^(1/2) is the square root of a, a^(1/3) is the cube root of a, and a^(3/2) is the square root of a³. Working out the root first usually makes the numbers smaller. Welders meet fractional powers in dimension calculations that involve square roots and cube roots.
Example 1: Calculate 16^(1/2).
16^(1/2) = √16
Answer: 4
Example 2: Find 27^(1/3).
27^(1/3) = ³√27
Answer: 3
Example 3: Evaluate 81^(3/4).
81^(3/4) = (⁴√81)³ = 3³ = 27
The fourth root is taken first, because 3⁴ = 81.
Answer: 27
Example 4: Simplify 125^(2/3).
125^(2/3) = (³√125)² = 5² = 25
Answer: 25
Example 5: Calculate 9^(3/2).
9^(3/2) = (√9)³ = 3³ = 27
Answer: 27
The laws of indices are the rules for combining powers of the same base. They save a great deal of working, and they are the basis of logarithms. In the laws below, a and b are non-zero numbers, and m and n are any indices.
The names show what is done to the indices in the first two laws: they are added when powers are multiplied, and subtracted when powers are divided.
When powers of the same base are multiplied, the indices are added: aᵐ × aⁿ = a^(m + n). The reason is that aᵐ has m factors of a, and aⁿ has n more, so the product has m + n factors in all. The rule applies only when the bases are the same.
Example 1: Simplify 3⁴ × 3².
3⁴ × 3² = 3^(4 + 2) = 3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 729
Answer: 729
Example 2: Simplify 2⁵ × 2⁻³.
2⁵ × 2⁻³ = 2^(5 − 3) = 2² = 4
Answer: 4
Example 3: Simplify x³ × x⁴ × x.
x³ × x⁴ × x¹ = x^(3 + 4 + 1) = x⁸
Answer: x⁸
When a power is divided by a power of the same base, the index of the divisor is subtracted from the index of the dividend: aᵐ ÷ aⁿ = a^(m − n). The result may be a negative index if n is larger than m.
Example 1: Simplify 5⁷ ÷ 5³.
5⁷ ÷ 5³ = 5^(7 − 3) = 5⁴ = 5 × 5 × 5 × 5 = 625
Answer: 625
Example 2: Simplify 2³ ÷ 2⁵.
2³ ÷ 2⁵ = 2^(3 − 5) = 2⁻² = 1 ÷ 2² = 1 ÷ 4
Answer: 1 ÷ 4 = 0.25
Example 3: Simplify y⁹ ÷ y⁶.
y⁹ ÷ y⁶ = y^(9 − 6) = y³
Answer: y³
When a quotient is raised to a power, the numerator and the denominator are each raised to that power: (a ÷ b)ⁿ = aⁿ ÷ bⁿ. This law is used when a fraction is raised to a power, such as a ratio of two sizes that is squared to compare areas.
Example 1: Simplify (6 ÷ 2)³.
(6 ÷ 2)³ = 6³ ÷ 2³ = 216 ÷ 8 = 27
Check: 6 ÷ 2 = 3, and 3³ = 27.
Answer: 27
Example 2: Simplify (2 ÷ 5)².
(2 ÷ 5)² = 2² ÷ 5² = 4 ÷ 25
Answer: 4 ÷ 25 = 0.16
When a power is raised to a further power, the indices are multiplied: (aᵐ)ⁿ = a^(m × n). When a product is raised to a power, each factor is raised to that power: (a × b)ⁿ = aⁿ × bⁿ.
Example 1: Evaluate (2³)⁴.
(2³)⁴ = 2^(3 × 4) = 2¹² = 4 096
Answer: 4 096
Example 2: Simplify (4 × 3)².
(4 × 3)² = 4² × 3² = 16 × 9 = 144
Check: 4 × 3 = 12, and 12² = 144.
Answer: 144
Example 3: Simplify (2x²)³.
(2x²)³ = 2³ × (x²)³ = 8 × x⁶ = 8x⁶
Answer: 8x⁶
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Create a free accountThis chapter covered the application of algebra beginning with indices algebra, where the concepts of power zero, negative powers, fractional powers, and the fundamental laws of indices including addition, subtraction, division, and multiplication of powers were explored. The importance of BODMAS was emphasized to ensure correct order of operations in algebraic expressions. Roots were examined in detail, focusing on square roots, cube roots, and general nth roots. The chapter then introduced logarithms and their laws, specifically the product, quotient, and power laws that simplify complex calculations. Practical use of scientific calculators was demonstrated, highlighting functions such as power on/off, mode settings for degrees, radians, and gradients, as well as clear, save, and shift operations. Methods for solving simultaneous equations involving up to three variables were discussed, including elimination, substitution, reduction, and graphical techniques. Finally, quadratic equations were analyzed through factorization, the quadratic formula, completing the square, and graphical solutions, providing a comprehensive foundation for algebraic problem-solving in engineering contexts.
Quadratic equations are frequently encountered in welding mathematics, especially when calculating weld bead widths, penetration depths, or optimizing parameters for joint strength. The standard form is:
$$ax^2 + bx + c = 0$$
There are several methods to solve quadratic equations: factorization, quadratic formula, completing the square, and graphical solutions.
Example 1: Solve the quadratic equation by factorization: \(x^2 - 7x + 10 = 0\).
Given: \(a = 1, b = -7, c = 10\)
Factorize:
Find two numbers that multiply to 10 and add to -7: -5 and -2
$$(x - 5)(x - 2) = 0$$
Set each to zero:
$$x - 5 = 0 \Rightarrow x = 5$$$$x - 2 = 0 \Rightarrow x = 2$$
Answer: \(x = 5\) or \(x = 2\)
Example 2: Solve \(2x^2 - 5x - 3 = 0\) using the quadratic formula.
Given: \(a = 2, b = -5, c = -3\)
Quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Calculate discriminant:
$$\Delta = (-5)^2 - 4 \times 2 \times (-3) = 25 + 24 = 49$$
Calculate roots:
$$x = \frac{-(-5) \pm \sqrt{49}}{2 \times 2} = \frac{5 \pm 7}{4}$$
First root:
$$x = \frac{5 + 7}{4} = \frac{12}{4} = 3$$
Second root:
$$x = \frac{5 - 7}{4} = \frac{-2}{4} = -0.5$$
Answer: \(x = 3\) or \(x = -0.5\)
Example 3: Complete the square for \(x^2 - 6x + 5 = 0\).
Given: \(a = 1, b = -6, c = 5\)
Rewrite:
$$x^2 - 6x = -5$$
Take half of -6, square it:
$$(-3)^2 = 9$$
Add 9 to both sides:
$$x^2 - 6x + 9 = -5 + 9$$$$ (x - 3)^2 = 4 $$
Take square root:
$$ x - 3 = \pm 2 $$
So:
$$ x = 3 + 2 = 5 $$$$ x = 3 - 2 = 1 $$
Answer: \(x = 5\) or \(x = 1\)
Example 4: Solve graphically: \(x^2 - 4x + 3 = 0\).
Plot the curve \(y = x^2 - 4x + 3\) and find where \(y = 0\).
Factoring:
$$(x - 1)(x - 3) = 0$$
So intersection points at \(x = 1\) and \(x = 3\).
Answer: \(x = 1\) or \(x = 3\)
Calculate \(5^0\), which represents the power zero rule often used when determining weld bead counts where no additional layers are applied. (2 marks)
Evaluate \(4^{-2}\) to find the reciprocal power relevant when calculating reduction factors in welding heat input. (3 marks)
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