If √(31 + 3√48) = a + b√3, what is the value of √(b2 - a2), correct to one…

Concept

When an expression of the form √(m + n√k) is written as a + b√k, square both sides and compare the rational part and the surd part separately.

This comparison gives equations such as a2 + k b2 = m and 2ab = n, which determine the coefficients before the requested expression is evaluated.

Application

1. Here √48 = 4√3, so 31 + 3√48 = 31 + 12√3.

2. Let √(31 + 12√3) = a + b√3. Squaring gives a2 + 3b2 + 2ab√3 = 31 + 12√3.

3. Compare the rational and surd parts: a2 + 3b2 = 31 and 2ab = 12, so ab = 6.

4. The positive pair a = 2, b = 3 satisfies a2 + 3b2 = 4 + 27 = 31 and ab = 6.

5. Then b2 - a2 = 9 - 4 = 5, so √(b2 - a2) = √5 ≈ 2.236..., which rounds to 2.2.

Cross-check

(2 + 3√3)2 = 4 + 27 + 12√3 = 31 + 12√3, so the coefficients reproduce the original radical.

Result

The required one-decimal value is 2.2.