Mathematics · Grade 11 · Year Summary Date: Top Journal

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PAPER 1 · ALGEBRAIC EXPRESSIONS EXPONENTS & SURDS

Learning outcomes · you must be able to

Apply all the laws of exponents, including negative and rational (fractional) exponents.
Convert freely between surd form and exponent form, and simplify surds to their simplest form.
Add, subtract, multiply and divide surds, and rationalise a denominator (including a binomial denominator, using a conjugate).
Factorise fully: common factor, difference of two squares, trinomials, grouping, and the sum and difference of two cubes.
Simplify algebraic fractions by factorising first, then cancelling common factors.

📖 KEY DEFINITION Exponent (power): the small raised number that shows how many times a base is multiplied by itself. Rational exponent: a fractional power, where a^m/n means the n^th root of a^m. Surd: an irrational root that cannot be simplified to a rational number, such as √2. Rationalising: rewriting a fraction so that no surd remains in the denominator. Conjugate: the pair (a + √b) and (a − √b), whose product is rational.

Core content · the laws of exponents

Law	Rule	Quick example
Product	a^m × aⁿ = a^m+n	x³ × x⁴ = x⁷
Quotient	a^m ÷ aⁿ = a^m−n	x⁷ ÷ x² = x⁵
Power of a power	(a^m)ⁿ = a^mn	(x²)³ = x⁶
Power of a product	(ab)ⁿ = aⁿbⁿ	(2x)³ = 8x³
Zero exponent	a⁰ = 1 (a ≠ 0)	(5y)⁰ = 1
Negative exponent	a⁻ⁿ = 1 ÷ aⁿ	2⁻³ = 1⁄8
Rational exponent	a^m/n = ⁿ√(a^m)	8^2/3 = 4

⚡ MUST MEMORISE

Surd facts: √(ab) = √a × √b and √(a÷b) = √a ÷ √b

To rationalise a single surd, multiply top and bottom by that surd. To rationalise a binomial denominator, multiply top and bottom by its conjugate, then use the difference of two squares.

Cubes: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²)

⭐ EXAM FAVOURITE Simplifying an expression where the same base appears in several terms that are added or subtracted, for example (5^x+1 + 5^x+2) ÷ (6 × 5^x). The trick is always the same: take out the smallest power as a common factor, then the base cancels and a plain number is left.

🔥 FREQUENTLY TESTED Rationalising a binomial denominator with a conjugate, simplifying surds by pulling out the largest perfect-square factor, and rewriting a rational exponent as a root (or the reverse) before simplifying.

Worked example 1 · common factor with a shared base

Q: Simplify (5^x+1 + 5^x+2) ÷ (6 × 5^x). (Invented for practice.)

Split each power: 5^x+1 = 5 × 5^x and 5^x+2 = 25 × 5^x

Numerator = 5 × 5^x + 25 × 5^x = 5^x(5 + 25) = 30 × 5^x

Expression = (30 × 5^x) ÷ (6 × 5^x) = 30 ÷ 6 = 5

∴ the expression simplifies to 5

Worked example 2 · rational exponents without a calculator

Q: Evaluate 8^2/3 + 16^1/2 − 25^1/2. (Invented for practice.)

8^2/3 = (³√8)² = 2² = 4

16^1/2 = √16 = 4

25^1/2 = √25 = 5

4 + 4 − 5 = 3

∴ the value is 3

Worked example 3 · rationalising a binomial denominator

Q: Rationalise the denominator of 3 ÷ (√5 − √2). (Invented for practice.)

Multiply top and bottom by the conjugate (√5 + √2):

= 3(√5 + √2) ÷ [(√5 − √2)(√5 + √2)]

Denominator (difference of two squares) = (√5)² − (√2)² = 5 − 2 = 3

= 3(√5 + √2) ÷ 3 = √5 + √2

∴ the simplified form is √5 + √2

Worked example 4 · simplifying and combining surds

Q: Simplify √48 − √27 + √12, and factorise 8x³ − 27. (Invented for practice.)

√48 = √(16 × 3) = 4√3 ; √27 = √(9 × 3) = 3√3 ; √12 = √(4 × 3) = 2√3

4√3 − 3√3 + 2√3 = 3√3

8x³ − 27 is a difference of two cubes with a = 2x, b = 3:

= (2x − 3)((2x)² + (2x)(3) + 3²) = (2x − 3)(4x² + 6x + 9)

∴ √48 − √27 + √12 = 3√3, and 8x³ − 27 = (2x − 3)(4x² + 6x + 9)

⚠️ COMMON MISTAKE Adding powers across a plus sign (2^x + 2^x is 2 × 2^x, NOT 4^x or 2^2x). Also: √(a + b) is NOT √a + √b, and a negative exponent flips the base to the other side of the fraction, it does NOT make the answer negative.

💡 EASY MARK "Write √50 in its simplest form" is a one-mark gift: pull out the largest perfect square, √50 = 5√2. Likewise any base to the power 0 is simply 1, no working needed.

Algebra rewards the patient: take the base apart, and the rule that was hiding inside shows itself.- on simplifying

Why this matters: exponents and surds open Paper 1, but they also sit underneath exponential graphs, compound interest and half the equations you will solve this year. The laws are short enough to learn in one sitting, and a learner who is shaky on them loses marks in three questions, not one.

🧠 ACTIVE RECALL

Simplify (x³)² × x⁴.(2)

The value of 16^3/4 is:(2)

√50 written in simplest surd form is:(2)

Rationalising 1 ÷ √3 gives:(2)

The value of 2⁻³ is:(2)

Factorising x² − 9 gives:(2)

Simplify fully: (2^x+3 − 2^x+1) ÷ 2^x.(3)

Rationalise the denominator and simplify: 4 ÷ (√6 + √2).(3)

Factorise fully: x³ − 8.(3)

Explain to a Grade 8: why is any number (except zero) to the power of 0 equal to 1?

ONE-MINUTE SUMMARY

Powers add across multiplication and subtract across division, but never combine across a plus or minus. A fractional power is a root: a^m/n is the n^th root of a^m. Simplify a surd by pulling out its largest perfect-square factor, and clear a surd from a denominator by multiplying by the conjugate. Factorise fully by trying common factor, difference of two squares, trinomials and, new this year, the sum and difference of two cubes.

IN THE EXAM

Exponent and surd simplification opens Paper 1 and feeds straight into the equations question that follows.
Show the common-factor line and the conjugate line in full: those method steps carry the marks even when the final answer is short.
Leave surd answers in exact form (5√2, not a rounded decimal) unless a decimal is specifically asked for.

🔒 EQUATIONS

🔒 PATTERNS

🔒 FUNCTIONS

🔒 FINANCE

🔒 PROBABILITY

🔒 ANALYTIC GEO

🔒

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