Joe Blessing Academia

14/08/2026

Simplifying surds

14/08/2026

📘 **JOE BLESSING ACADEMIA** — Math Note 🧮

**TOPIC: SURDS**

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**A. WHAT IS A SURD?**

A surd is an irrational root that cannot be simplified to remove the root — e.g. √2, √3, √5, ∛7.

Note: √4 = 2 is **not** a surd (it simplifies to a whole number).

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**B. LAWS OF SURDS**

1️⃣ √a × √b = √(ab)
2️⃣ √a ÷ √b = √(a/b)
3️⃣ a√c ± b√c = (a±b)√c (like surds only)
4️⃣ √a × √a = a
5️⃣ (√a)² = a

**Example:**
Simplify: √8 + √18
= √(4×2) + √(9×2)
= 2√2 + 3√2
= 5√2

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**C. RATIONALIZING THE DENOMINATOR**

Rule: Multiply top and bottom by the **conjugate** of the denominator.

**Example 1:**
Rationalize: 1/√3
= (1×√3)/(√3×√3)
= √3/3

**Example 2:**
Rationalize: 1/(2+√3)
Multiply by conjugate (2−√3):
= (2−√3) / [(2+√3)(2−√3)]
= (2−√3) / (4−3)
= 2−√3

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**D. MULTIPLYING SURDS**

**Example:**
(√5 + √2)(√5 − √2)
= (√5)² − (√2)²
= 5 − 2 = 3

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**E. QUICK TIP**

Always simplify a surd first by breaking it into a **perfect square factor** × another factor.
E.g. √50 = √(25×2) = 5√2

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✏️ **Practice Question:**
Rationalize: 3/(√5 − √2)

💬 Drop your working in the comments!

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Want the next one to be Quadratic Equations, Simultaneous Equations, or a Surds/Indices/Logs mixed exam-style worksheet?

13/08/2026

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13/08/2026

Solve for the value of x

13/08/2026

📘 **JOE BLESSING ACADEMIA** — Math Note 🧮

**TOPIC: INDICES AND LOGARITHMS**

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**A. LAWS OF INDICES**

If a and b are bases, m and n are exponents:

1️⃣ aᵐ × aⁿ = aᵐ⁺ⁿ
2️⃣ aᵐ ÷ aⁿ = aᵐ⁻ⁿ
3️⃣ (aᵐ)ⁿ = aᵐⁿ
4️⃣ a⁰ = 1 (a ≠ 0)
5️⃣ a⁻ⁿ = 1/aⁿ
6️⃣ a^(1/n) = ⁿ√a
7️⃣ a^(m/n) = ⁿ√(aᵐ)
8️⃣ (ab)ⁿ = aⁿbⁿ
9️⃣ (a/b)ⁿ = aⁿ/bⁿ

**Example:**
Simplify: 2³ × 2⁴ ÷ 2²
= 2^(3+4-2) = 2⁵ = 32

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**B. LAWS OF LOGARITHMS**

If logₐN = x, then aˣ = N

1️⃣ logₐ(MN) = logₐM + logₐN
2️⃣ logₐ(M/N) = logₐM − logₐN
3️⃣ logₐ(Mⁿ) = n·logₐM
4️⃣ logₐa = 1
5️⃣ logₐ1 = 0
6️⃣ Change of base: logₐN = logᵦN / logᵦa

**Example:**
Evaluate: log₂8 + log₂4
= log₂(8×4) = log₂32 = 5
(since 2⁵ = 32)

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**C. RELATIONSHIP BETWEEN INDICES & LOGARITHMS**

If aˣ = N, then x = logₐN

This means logarithms are simply the **inverse** of indices — logs answer the question "what power?" while indices answer "what value?"

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✏️ **Practice Question:**
Solve for x: log₃(x) = 4

💬 Drop your answer in the comments!

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13/08/2026

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13/08/2026

Simplify this equation and solve for x

12/08/2026

12/08/2026

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11/08/2026

📐 JOE BLESSING ACADEMIA
🎓 SSS MATHEMATICS | INDICES

🔷 LESSON 5: ADVANCED SIMPLIFICATION OF INDICES

At this level, examination questions may combine positive, negative and algebraic indices.

✍️ WORKED EXAMPLE 1

Simplify:

x⁷y⁻² ÷ x³y⁻⁵

Using the quotient law:

= x⁷⁻³ y⁻²⁻⁽⁻⁵⁾

= x⁴y³

✅ Answer: x⁴y³

✍️ WORKED EXAMPLE 2

Simplify:

(2a²b⁻¹)³

Apply the power to each factor:

= 2³ × a⁶ × b⁻³

= 8a⁶/b³

✅ Answer: 8a⁶/b³

✍️ WORKED EXAMPLE 3

Simplify:

(x³y²)² ÷ x⁴y

First expand:

= x⁶y⁴ ÷ x⁴y

Using the quotient law:

= x⁶⁻⁴y⁴⁻¹

= x²y³

✅ Answer: x²y³

🎯 EXAM STRATEGY

When solving complex indices:

1️⃣ Expand powers first.
2️⃣ Add indices when multiplying like bases.
3️⃣ Subtract indices when dividing like bases.
4️⃣ Convert negative indices to positive indices where necessary.

🧠 CHALLENGE QUESTION

Simplify completely:

(2x⁻²y³)² ÷ 4x⁻⁵y

👇 Solve it and drop your answer in the comments.

📌 NEXT LESSON:
Fractional Indices and Surds

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