If x = √198 × √550 and y = ∛99 × ∛363, then 1/x + 1/y is equal to:

Concept

For radicals of the same index the product rule applies: the square root of a times the square root of b equals the square root of (ab), and the cube root of a times the cube root of b equals the cube root of (ab). The merged radicand is then simplified by prime factorisation, because a perfect square leaves a whole number under a square root and a perfect cube does the same under a cube root.

A recurring decimal is turned into a fraction by a fixed rule: a block of n digits that recurs immediately after the decimal point equals that block divided by n nines, while k non-recurring digits followed by a recurring block equal (the whole digit string minus the non-recurring part) divided by n nines followed by k zeros.

Application

1. x = √198 × √550 = √(198 × 550) = √108900.

2. 198 = 2 × 32 × 11 and 550 = 2 × 52 × 11, so 198 × 550 = 22 × 32 × 52 × 112. Taking the square root of each factor gives x = 2 × 3 × 5 × 11 = 330.

3. y = ∛99 × ∛363 = ∛(99 × 363) = ∛35937.

4. 99 = 32 × 11 and 363 = 3 × 112, so 99 × 363 = 33 × 113. Taking the cube root of each factor gives y = 3 × 11 = 33.

5. 1/x + 1/y = 1/330 + 1/33. Writing both over 330 gives 1/330 + 10/330 = 11/330, which reduces to 1/30.

6. Expressing 1/30 as a decimal gives 1/30 = 0.03333… , that is 0.03 with a bar over the digit 3.

Cross-check

• 3302 = 108900 and 333 = 35937, so both radicals were simplified correctly.

• Reversing the decimal rule: 0.03 with a bar over 3 equals (3 − 0)/90 = 3/90 = 1/30, which matches the sum obtained above.

• The remaining listed values work out to 01/99 = 1/99, 33/99 = 1/3 and 1/9, each different from 1/30.