KCSE 2025 Maths P2 Q3 — Rationalising Surds to the Form a + b root 2

KCSE 2025 Form 4 Surds

Published

The Question

“Find the value of a and b for which 7 root 2 divided by (5 minus 3 root 2) equals a plus b root 2.”

1

Multiply by the conjugate of the denominator

To clear the surd from the bottom, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (5 minus 3 root 2) is (5 plus 3 root 2), the same terms with the middle sign flipped. Because this fraction equals 1, it changes the form of the expression but not its value.

725−32×5+325+32\frac{7\sqrt{2}}{5 - 3\sqrt{2}} \times \frac{5 + 3\sqrt{2}}{5 + 3\sqrt{2}}
2

Expand the numerator

Multiply 7 root 2 across both terms of the conjugate. The first product keeps the root, while in the second the two roots combine because root 2 times root 2 is 2, turning it into a whole number.

72(5+32)=352+21×27\sqrt{2}(5 + 3\sqrt{2}) = 35\sqrt{2} + 21 \times 2
=352+42= 35\sqrt{2} + 42
3

Expand the denominator as a difference of two squares

Multiplying a surd expression by its conjugate gives the difference of two squares, so the cross terms cancel and the root disappears. Square the 5 and square the 3 root 2 (remembering to square the root as well), then subtract.

(5−32)(5+32)=52−(32)2(5 - 3\sqrt{2})(5 + 3\sqrt{2}) = 5^{2} - (3\sqrt{2})^{2}
=25−9×2=25−18=7= 25 - 9 \times 2 = 25 - 18 = 7
4

Divide each term and read off a and b

Now the fraction has a whole-number denominator. Divide each term of the numerator by 7 to simplify. Comparing the result with the required form a plus b root 2 lets you read off the two whole numbers directly.

352+427=6+52\frac{35\sqrt{2} + 42}{7} = 6 + 5\sqrt{2}
a=6,b=5a = 6, \quad b = 5

Final Result

a = 6 and b = 5, so the expression simplifies to 6 + 5 root 2.

Why this method works

A surd in the denominator is removed by exploiting the difference of two squares: multiplying (5 minus 3 root 2) by its conjugate (5 plus 3 root 2) squares each term, and squaring a root turns it into a rational number, so all the roots on the bottom vanish. Since the conjugate over itself equals 1, the value of the expression is unchanged while its form becomes a rational denominator, letting you split it into a rational part and a surd part that match a and b.

✓

Multiply back: (6 + 5 root 2)(5 - 3 root 2) = 30 - 18 root 2 + 25 root 2 - 30 = 7 root 2, which is the original numerator, confirming the answer.