ARCH-M1-02: Surd Rationalisation, Difference-of-Squares & Laws of Indices

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-M1-02: Why the 125s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~125s (Blows 90s Clock)

Expand

(2\sqrt{3} - \sqrt{2})(2\sqrt{3} - \sqrt{2}) = 4(3) - 2\sqrt{6} - 2\sqrt{6} + 2 = 14 - 4\sqrt{6}

. Comparing with

a - b\sqrt{6}

gives

a = 14

and

b = 4

, so

a + b = 18

(~105s).

The Fast ShortcutTarget: 35s (Saves 90s)

Line 1 (Binomial Square):

(2\sqrt{3})^2 + (\sqrt{2})^2 = 12 + 2 = 14 = a

, and cross-term coefficient

b = 2 \times 2 = 4

.
Line 2 (Sum):

a + b = 14 + 4 = 18

(25s).

Where Students Lose Time: In 17 past UAT-UK questions (7.56% of Mathematics 1), candidates lose 90s by subtracting the squared surd term in expansion or multiplying by (a - sqrt(b)) instead of conjugate (a + sqrt(b)). Instead, apply M2 with Maths 1 skill M2.5.

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