Porisma

Worked solution

Simplify a surd product and rationalise a denominator

Simplify the following: (a) 3×27\sqrt3\times\sqrt{27}; (b) 13−2\dfrac1{3-\sqrt2}.

CJC · O Level revision · Example 6

(a) Extract the square factor

Answer

99.

Write 27 as a cube of 3

327=333=3 (33)=9\sqrt3\sqrt{27}=\sqrt3\sqrt{3^3}=\sqrt3\,(3\sqrt3)=9

The source extracts a factor 3 from the square root, then multiplies the equal surds.

How did this part go?

(b) Multiply by the conjugate factor

Answer

3+27\dfrac{3+\sqrt2}{7}.

Multiply by one in a useful form

13−2=13−23+23+2\frac1{3-\sqrt2}=\frac1{3-\sqrt2}\frac{3+\sqrt2}{3+\sqrt2}

The factor 3+23+\sqrt2 is nonzero, so multiplying numerator and denominator by it preserves the value.

Use the difference of squares

3+232−(2)2=3+27\frac{3+\sqrt2}{3^2-(\sqrt2)^2}=\frac{3+\sqrt2}{7}

This is the printed rationalisation: the denominator becomes 9−2=7.

How did this part go?