How to Solve: Algebraic Expressions and Surds (Indices, Rationalising, Partial Fractions)

How to Solve: Algebraic Expressions and Surds (Indices, Rationalising, Partial Fractions)

Introduction

"Mastering surds and indices doesn’t just get you marks—it unlocks the hardest algebra questions on your exam, worth up to 15% of your total score. Miss this, and you’re leaving easy points on the table."

(Pause for emphasis) "Today, you’ll learn the exact steps to simplify, rationalise, and break down expressions—so you can solve any question in under 2 minutes."

What You Need To Know First

Before diving in, make sure you’re solid on: 1. Basic index laws (e.g., (a^m \times a^n = a^{m+n})). 2. Fraction arithmetic (adding, subtracting, multiplying fractions). 3. Expanding brackets (e.g., ((a + b)(c + d) = ac + ad + bc + bd)).

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

1. Index Laws (MEMORISE THIS)

2. Rationalising the Denominator (MEMORISE THIS)

• Single surd: Multiply numerator and denominator by the surd. (\frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a})
• Binomial surd: Multiply by the conjugate. (\frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b})

3. Partial Fractions (Given on exam sheet for A-Level, but know how to use it)

For (\frac{P(x)}{Q(x)}) where (Q(x)) is a product of linear factors: - Distinct linear factors: (\frac{A}{x + a} + \frac{B}{x + b}) - Repeated linear factor: (\frac{A}{x + a} + \frac{B}{(x + a)^2})

Step-by-Step Method

Part 1: Simplifying Surds & Indices

Step 1: Identify the largest perfect square factor of the number under the root. Step 2: Split the surd into two parts: (\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}). Step 3: Simplify the perfect square part. Step 4: Multiply the simplified parts back together.

Example: Simplify (\sqrt{50}). 1. Largest perfect square factor of 50 is 25. 2. (\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}). 3. (\sqrt{25} = 5). 4. Final answer: (5\sqrt{2}).

Part 2: Rationalising the Denominator

Step 1: Check if the denominator is a single surd or a binomial surd. Step 2:
- If single surd, multiply numerator and denominator by the surd.
- If binomial surd, multiply numerator and denominator by the conjugate. Step 3: Expand the numerator and denominator. Step 4: Simplify the denominator (it should no longer have a surd). Step 5: Simplify the fraction if possible.

Example: Rationalise (\frac{3}{2 + \sqrt{5}}). 1. Denominator is a binomial surd. 2. Multiply by conjugate: (\frac{3}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}}). 3. Numerator: (3(2 - \sqrt{5}) = 6 - 3\sqrt{5}).
Denominator: ((2 + \sqrt{5})(2 - \sqrt{5}) = 4 - (\sqrt{5})^2 = 4 - 5 = -1). 4. Simplified: (\frac{6 - 3\sqrt{5}}{-1} = -6 + 3\sqrt{5}). 5. Final answer: (3\sqrt{5} - 6).

Part 3: Partial Fractions (A-Level Only)

Step 1: Factorise the denominator completely. Step 2: Write the general form of partial fractions based on the factors. Step 3: Multiply both sides by the denominator to eliminate fractions. Step 4: Solve for the unknowns (A, B, etc.) by:
- Substituting values of (x) that make terms zero.
- Or expanding and equating coefficients. Step 5: Write the final answer as separate fractions.

Example: Express (\frac{5x + 3}{(x + 1)(x - 2)}) in partial fractions. 1. Denominator is already factorised. 2. General form: (\frac{A}{x + 1} + \frac{B}{x - 2}). 3. Multiply both sides by ((x + 1)(x - 2)):
(5x + 3 = A(x - 2) + B(x + 1)). 4. Solve for A and B:
- Let (x = -1): (5(-1) + 3 = A(-3) + B(0) \Rightarrow -2 = -3A \Rightarrow A = \frac{2}{3}).
- Let (x = 2): (5(2) + 3 = A(0) + B(3) \Rightarrow 13 = 3B \Rightarrow B = \frac{13}{3}). 5. Final answer: (\frac{2/3}{x + 1} + \frac{13/3}{x - 2}).

Worked Examples

Example 1 – Basic: Simplify (\sqrt{72})

Step 1: Largest perfect square factor of 72 is 36. Step 2: (\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}). Step 3: (\sqrt{36} = 6). Step 4: Final answer: (6\sqrt{2}).

What we did and why: We split the surd into a perfect square and another number to simplify it. This makes the expression cleaner and easier to work with in later steps.

Example 2 – Medium: Rationalise (\frac{4}{\sqrt{7} - 2})

Step 1: Denominator is a binomial surd. Step 2: Multiply by conjugate: (\frac{4}{\sqrt{7} - 2} \times \frac{\sqrt{7} + 2}{\sqrt{7} + 2}). Step 3: Numerator: (4(\sqrt{7} + 2) = 4\sqrt{7} + 8).
Denominator: ((\sqrt{7})^2 - (2)^2 = 7 - 4 = 3). Step 4: Simplified: (\frac{4\sqrt{7} + 8}{3}). Step 5: Final answer: (\frac{4\sqrt{7}}{3} + \frac{8}{3}).

What we did and why: We used the conjugate to eliminate the surd in the denominator. This is a standard technique for rationalising binomial surds.

Example 3 – Exam-Style: Express (\frac{2x^2 + 5x - 3}{(x + 1)(x - 1)}) in partial fractions.

Step 1: Denominator is ((x + 1)(x - 1) = x^2 - 1) (difference of squares). Step 2: Since the numerator is degree 2 and denominator is degree 2, we need a linear term in the numerator:
(\frac{2x^2 + 5x - 3}{(x + 1)(x - 1)} = A + \frac{B}{x + 1} + \frac{C}{x - 1}). Step 3: Multiply both sides by ((x + 1)(x - 1)):
(2x^2 + 5x - 3 = A(x + 1)(x - 1) + B(x - 1) + C(x + 1)). Step 4: Expand and simplify:
(2x^2 + 5x - 3 = A(x^2 - 1) + Bx - B + Cx + C).
(2x^2 + 5x - 3 = Ax^2 + (B + C)x + (-A - B + C)). Step 5: Equate coefficients:
- (x^2): (A = 2).
- (x): (B + C = 5).
- Constants: (-A - B + C = -3 \Rightarrow -2 - B + C = -3 \Rightarrow -B + C = -1). Step 6: Solve the system:
(B + C = 5)
(-B + C = -1)
Add the equations: (2C = 4 \Rightarrow C = 2).
Substitute: (B + 2 = 5 \Rightarrow B = 3). Step 7: Final answer: (2 + \frac{3}{x + 1} + \frac{2}{x - 1}).

What we did and why: We recognised that the numerator’s degree was equal to the denominator’s, so we included a constant term (A). Then, we used coefficient matching to solve for the unknowns.

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute checklist for surds and indices:

1. Simplify surds by splitting them into perfect squares. (\sqrt{50} = 5\sqrt{2}).
2. Rationalise denominators by multiplying by the surd or its conjugate. No surds left downstairs!
3. Index laws are your best friend—add exponents when multiplying, subtract when dividing.
4. Partial fractions? Factorise the denominator, write the general form, then solve for A and B.
5. Watch for traps: hidden surds, repeated roots, and disguised indices.

Now go smash those exam questions—you’ve got this!