The third proportional to a³ + b³ and a² + ab + b², when a = 2 and b = 3, is:

(correct to 2 decimal places)

Given:

a = 2

b = 3

Concept:

We need to find the third proportional to a³ + b³ and a² + ab + b².

Solution:

⇒ Substitute a and b into the two expressions to get the first and second numbers.

⇒ First number = a³ + b³ = 2³ + 3³ = 8 + 27 = 35

⇒ Second number = a2 + ab + b2 = 22 + 2*3 + 32 = 4 + 6 + 9 = 19

⇒ The third proportional (T) to two numbers (x and y) is given by the formula T = (y²)/x

So, substituting the first and second numbers:

⇒ T = (19²)/35 = 10.31

Therefore, the third proportional to a³ + b³ and a² + ab + b², when a = 2 and b = 3, is approximately 10.31 (correct to two decimal places).