The laws of indices, fractional and negative powers, and solving exponential equations by matching bases or by substitution to a quadratic.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
If 2ˣ = 32 then x = 5 — but how do you solve 2²ˣ⁺¹ − 3(2ˣ) − 2 = 0? The trick is to see the quadratic hiding inside.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Change the base of y = a^x.
Copy these into your notebook in your own words — that is how they stick.
aᵐaⁿ = a^{m+n}; aᵐ/aⁿ = a^{m−n}; (aᵐ)ⁿ = a^{mn}; a⁰ = 1; a^{−n} = 1/aⁿ; a^{m/n} = (ⁿ√a)ᵐ.
Write each side as a power of the same base, then equate the indices.
u = aˣ; solve for u; reject u ≤ 0; then solve aˣ = u.
2^{x+1} = 2·2ˣ and 2^{2x} = (2ˣ)²: rewrite before substituting.
aˣ > 0 for every real x (a > 0).
a^{1/2} = √a; a^{1/3} = ∛a.
Take logarithms of both sides (A9).
Substitute x back into the original equation.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (27 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [15 marks]
Answer all parts.
(a) Solve 2^(2x+1) − 3(2ˣ) − 2 = 0. [5]
(b) Solve 3^(2x) − 4(3ˣ) + 3 = 0. [4]
(c) Solve 9^x = 1/27. [3]
(d) Show that (1/8)^(−2/3) = 4. [3]
Question 2 [12 marks]
The population of bacteria in a dish after t hours is N = 500 × 2^t.
(a) Find the number of bacteria at the start and after 3 hours. [3]
(b) Find the value of t when N = 32 000. [3]
(c) Find, to 2 decimal places, the time taken for the population to reach 10 000. [4]
(d) Explain why 2^t = −4 has no solution. [2]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.