Simplify: (125)1/3 × (4)−1/2 + 4(216)1/3(10)0 ÷ (42/√64)

Concept: BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) fixes the order in which a mixed expression must be simplified. Orders covers exponents and roots — a fractional exponent a1/n means the nth root of a, a negative exponent a−n means 1/an, and any nonzero base raised to the power 0 equals 1. Only after every bracket and every exponent/root is resolved do the multiplications and divisions run left to right, followed by the additions and subtractions.

Application (evaluating the given expression):

1. Resolve every exponent and root: (125)1/3 = 5, since 53 = 125; (4)−1/2 = 1/√4 = 1/2; (216)1/3 = 6, since 63 = 216; (10)0 = 1.

2. Resolve the bracket in the divisor: 42 ÷ √64 = 16 ÷ 8 = 2.

3. Substitute these values back into the expression: 5 × 1/2 + 4 × 6 × 1 ÷ 2.

4. Apply multiplication and division left to right: 5 × 1/2 = 5/2, and 4 × 6 × 1 ÷ 2 = 24 ÷ 2 = 12.

5. Apply the final addition: 5/2 + 12 = 5/2 + 24/2 = 29/2.

Cross-check:

Writing both terms over the common denominator 2 before adding — 5/2 and 24/2 — reproduces the same 29/2 regardless of the order in which the two product terms were simplified, confirming the result.