3.1 Evaluating Algebraic Expressions & Brackets

Algebraic Expressions in Aviation Maintenance

In aviation maintenance, technical manuals and regulations frequently express physical laws as algebraic expressions. An algebraic expression is a mathematical phrase containing numbers, variables (letters representing physical quantities like velocity $v$, density $\rho$, or force $F$), and operators ($+$, $-$, $\times$, $\div$). For instance, the aerodynamic lift generated by a wing is modeled by the expression:

$L = \frac{1}{2}\rho v^2 S C_L$

Here, $\rho$ represents the air density, $v$ is the true airspeed, $S$ is the wing planform area, and $C_L$ is the lift coefficient. In this expression, $\frac{1}{2}$ is a constant, while $\rho$, $v$, $S$, and $C_L$ are variables. A term is a single number, a variable, or numbers and variables multiplied together. In the expression $3x^2 - 5x + 2$, there are three terms: $3x^2$, $-5x$, and $2$. The number multiplying a variable is the coefficient (e.g., $3$ is the coefficient of $x^2$).

Order of Operations (PEMDAS/BODMAS) in Algebra

When evaluating algebraic expressions, you must strictly adhere to the order of operations. In EASA examinations, questions often test your ability to correctly sequence operations when brackets and negative signs are involved. The priority rules are defined by the acronyms PEMDAS or BODMAS:

1. Parentheses / Brackets: Perform operations inside the innermost grouping symbols first.
2. Exponents / Orders: Evaluate powers, indices, and roots (e.g., $x^2$, $\sqrt{y}$).
3. Multiplication and Division: Perform these operations from left to right as they appear. They hold equal priority.
4. Addition and Subtraction: Perform these operations from left to right as they appear. They also hold equal priority.

Handling Negative Signs and Brackets

A common source of error on the EASA Part-66 exam is the distribution of negative signs. A negative sign immediately preceding a bracket acts as a multiplier of $-1$. For example:

$-(x - y) = -1 \cdot (x - y) = -x + y$

Similarly, if we have nested brackets:

$3a - [2b - (c - d)] = 3a - [2b - c + d] = 3a - 2b + c - d$

Always resolve nested brackets from the inside out.

Step-by-Step Nested Simplification Example

Let's simplify the following expression before substituting any values: $E = 5x - [2(3x - y) - 3(x + 2y)]$

1. First, expand the internal terms inside the square brackets by distributing the multipliers: $2(3x - y) = 6x - 2y$ $-3(x + 2y) = -3x - 6y$
2. Substitute these expanded terms back into the square brackets: $E = 5x - [6x - 2y - 3x - 6y]$
3. Combine like terms inside the square brackets: $6x - 3x = 3x$ $-2y - 6y = -8y$ So: $E = 5x - [3x - 8y]$
4. Distribute the negative sign outside the square brackets: $E = 5x - 3x + 8y$
5. Combine final terms: $E = 2x + 8y$

If we evaluate this at $x = 4$ and $y = -1$: $E = 2(4) + 8(-1) = 8 - 8 = 0$

Using this systematic approach prevents errors related to sign changes and operations order.

Evaluating Expressions by Substitution

To evaluate an expression means to find its numerical value by replacing the variables with specific numbers. Let us examine two typical EASA-style aviation problems:

Worked Example 1: Aerodynamic Lift Calculation

Problem: Calculate the total lift ($L$) produced by an aircraft wing under the following conditions:

• Air density ($\rho$) = $1.225\text{ kg/m}^3$ (standard sea-level density)
• True airspeed ($v$) = $80\text{ m/s}$
• Wing surface area ($S$) = $25\text{ m}^2$
• Lift coefficient ($C_L$) = $1.4$ (dimensionless)

Formula: $L = \frac{1}{2}\rho v^2 S C_L$

Step-by-step Solution:

1. Substitute the numerical values into the expression: $L = \frac{1}{2} \cdot 1.225 \cdot 80^2 \cdot 25 \cdot 1.4$
2. According to PEMDAS/BODMAS, evaluate the exponent first ($80^2 = 6400$): $L = 0.5 \cdot 1.225 \cdot 6400 \cdot 25 \cdot 1.4$
3. Perform the multiplications from left to right:
   • $0.5 \cdot 1.225 = 0.6125$
   • $0.6125 \cdot 6400 = 3920$
   • $3920 \cdot 25 = 98000$
   • $98000 \cdot 1.4 = 137200\text{ N}$ (or $137.2\text{ kN}$)

Worked Example 2: Aircraft Engine Power

Problem: The brake horsepower (BHP) of a piston engine can be evaluated using: $\text{BHP} = \frac{P \cdot L \cdot A \cdot N \cdot K}{33000}$ Evaluate the BHP if:

• Mean effective pressure ($P$) = $125\text{ psi}$
• Stroke length ($L$) = $0.5\text{ feet}$
• Piston area ($A$) = $15\text{ in}^2$
• Engine speed ($N$) = $2400\text{ RPM}$
• Number of cylinders ($K$) = $6$

Solution:

1. Substitute the values and multiply the terms in the numerator: $P \cdot L \cdot A \cdot N \cdot K = 125 \cdot 0.5 \cdot 15 \cdot 2400 \cdot 6$ $125 \cdot 0.5 = 62.5$ $62.5 \cdot 15 = 937.5$ $937.5 \cdot 2400 = 2,250,000$ $2,250,000 \cdot 6 = 13,500,000$
2. Divide by the denominator: $\text{BHP} = \frac{13,500,000}{33000} \approx 409.09\text{ HP}$

Expanding Brackets (Distributive Law)

Expanding brackets is the process of removing parentheses by multiplying the term outside the brackets by every term inside.

Single Brackets

The basic rule is the distributive law: $a(b + c) = ab + ac$

Example: Expand $3x(2x - 4y)$. $3x(2x - 4y) = (3x \cdot 2x) + (3x \cdot -4y) = 6x^2 - 12xy$

Double Brackets

When multiplying two binomials, use the FOIL method (First, Outer, Inner, Last): $(a + b)(c + d) = ac + ad + bc + bd$

Example: Expand $(x - 3)(2x + 5)$.

• First: $x \cdot 2x = 2x^2$
• Outer: $x \cdot 5 = 5x$
• Inner: $-3 \cdot 2x = -6x$
• Last: $-3 \cdot 5 = -15$

Combine like terms ($5x - 6x = -x$): $(x - 3)(2x + 5) = 2x^2 - x - 15$

Special Algebraic Identities

Three key identities frequently appear in EASA examination simplifications:

1. Perfect Square (Sum): $(a + b)^2 = a^2 + 2ab + b^2$
2. Perfect Square (Difference): $(a - b)^2 = a^2 - 2ab + b^2$
3. Difference of Two Squares: $(a + b)(a - b) = a^2 - b^2$

Caution: A common mistake is writing $(a - b)^2 = a^2 - b^2$. Always remember the middle term ($\pm 2ab$).

Factoring Simple Algebraic Expressions

Factoring is the inverse of expanding. It involves breaking an expression down into a product of simpler factors.

1. Extracting the Greatest Common Factor (GCF)

Look for the largest factor common to all terms and write it outside the brackets. Example: Factor the expression for the total surface area of a cylindrical hydraulic actuator piston, given by $2\pi r^2 + 2\pi rh$. Here, both terms share the factor $2\pi r$. Extracting it gives: $2\pi r^2 + 2\pi rh = 2\pi r(r + h)$

2. Factoring the Difference of Two Squares

Any expression of the form $a^2 - b^2$ factors into: $a^2 - b^2 = (a - b)(a + b)$

This is highly useful for calculating the cross-sectional area of hollow tubular components, such as push-rods or aircraft hydraulic lines. Let the outer radius of a tube be $R$ and the inner radius be $r$. The cross-sectional area is: $A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2) = \pi(R - r)(R + r)$

If $R = 12\text{ mm}$ and $r = 10\text{ mm}$: $A = \pi(12 - 10)(12 + 10) = \pi(2)(22) = 44\pi \approx 138.23\text{ mm}^2$

This factoring method permits rapid mental estimation before formal computation.