Resolve: √17.64 + √70.56 ÷ 2 = ?

CONCEPT: BODMAS fixes the order in which operations are evaluated — Brackets, Of, Division, Multiplication, Addition, Subtraction — so division must be carried out before addition. The square root of a decimal can be found by writing it as a product of prime factors (including the decimal's own factors of 0.1), pairing identical factors, and taking one factor from each pair out of the root.

APPLICATION:

1. Factorize 17.64 = 2 × 2 × 3 × 3 × 7 × 7 × 0.1 × 0.1. Taking one factor from each pair gives √17.64 = 2 × 3 × 7 × 0.1 = 4.2.

2. Factorize 70.56 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7 × 0.1 × 0.1. Taking one factor from each pair gives √70.56 = 2 × 2 × 3 × 7 × 0.1 = 8.4.

3. By BODMAS, division is applied before addition: 8.4 ÷ 2 = 4.2.

4. Add this to the first square root: 4.2 + 4.2 = 8.4.

CROSS-CHECK: 70.56 = 4 × 17.64, so √70.56 = 2 × √17.64 = 2 × 4.2 = 8.4, confirming step 2 independently. Squaring back also confirms both roots: 4.22 = 17.64 and 8.42 = 70.56.

So the expression evaluates to 8.4.