Grade 11 Exponents and Surds

Key concepts

• Rational exponents: a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
• Simplifying surds: √(ab) = √a × √b
• Rationalising the denominator: 1/√2 = √2/2
• Exponential equations by making the bases equal or by factorising
• Surd equations: isolate the surd, square both sides, then check

Worked examples

Simplify: 8^(2/3)

8^(1/3) = 2 (the cube root of 8), and 2² = 4. So 8^(2/3) = 4.

Simplify: √48 − 2√12

√48 = √16 × √3 = 4√3 and 2√12 = 2 × 2√3 = 4√3. So √48 − 2√12 = 0.

Solve: √(x + 7) = x − 5

Square both sides: x + 7 = x² − 10x + 25, so x² − 11x + 18 = 0 and (x − 9)(x − 2) = 0. Check: x = 9 gives √16 = 4 = 9 − 5 ✓; x = 2 gives √9 = 3 but 2 − 5 = −3 ✗. So x = 9 only.

Solve: 2ˣ⁺¹ + 2ˣ = 12

Factorise: 2ˣ(2 + 1) = 12, so 3 · 2ˣ = 12 and 2ˣ = 4 = 2². So x = 2.

Rationalise the denominator: 6/√3

Multiply the top and bottom by √3: 6√3/(√3 × √3) = 6√3/3 = 2√3. The value doesn't change, because you multiplied by √3/√3 = 1.

Common mistakes

• Not checking surd equations, so a false root is kept.
• Writing √(a + b) = √a + √b, which is false.
• Mixing up a^(2/3) as (a²)^(1/3) versus 2/3 × a.
• Trying to apply exponent laws to a sum before factorising.

Exam tips

• Write numbers as powers of primes before applying any law.
• Always write 'Check:' and substitute both answers in a surd equation; examiners award a mark for rejecting the false root.
• Leave answers in simplest surd form unless a decimal is asked for.

Frequently asked questions

What is a surd?

An irrational root, such as √2 or ∛5, that can't be simplified to a rational number. √9 = 3 is not a surd.

Why do surd equations give false roots?

Squaring both sides can turn a false statement into a true one: −3 ≠ 3, but (−3)² = 3². So a solution of the squared equation may not satisfy the original one.

Are logarithms in Grade 11?

No. Logarithms are introduced in Grade 12 as the inverse of the exponential function.