For what value of A, will the expression (13.56 × 13.56 + 13.56 × A + 0.04 × 0.04) be a perfect square?

Given:

Expression: (13.56 × 13.56 + 13.56 × A + 0.04 × 0.04)

Formula Used:

The algebraic identity for a perfect square is (a + b)² = a² + 2ab + b²

Calculation:

Let's compare the given expression with the perfect square identity:

a² = 13.56 × 13.56 = (13.56)² => a = 13.56

b² = 0.04 × 0.04 = (0.04)² => b = 0.04

The middle term in the perfect square identity is 2ab.

Comparing this with the middle term of the given expression, 13.56 × A:

2ab = 13.56 × A

Substitute the values of a and b:

2 × 13.56 × 0.04 = 13.56 × A

0.08 × 13.56 = 13.56 × A

A = 0.08

So, the expression is equal to (13.56 + 0.04)² = (13.60)², which is a perfect square.

For A = 0.08, the expression will be a perfect square.