What does (a + √b)² equal?

To determine what ( (a + \sqrt{b})^2 ) equals, we can apply the expansion formula for a binomial squared, which is given by ( (x + y)^2 = x^2 + 2xy + y^2 ).

In this case, ( x ) is ( a ) and ( y ) is ( \sqrt{b} ). Following the formula:

1. The square of the first term ( a ) is ( a^2 ).
2. The product of the two terms ( 2xy ) becomes ( 2 \cdot a \cdot \sqrt{b} ), which simplifies to ( 2a\sqrt{b} ).
3. The square of the second term ( \sqrt{b} ) is ( (\sqrt{b})^2 ), resulting in ( b ).

Putting this all together, we have:

[ (a + \sqrt{b})^2 = a^2 + 2a\sqrt{b} + b ]

Thus, ( (a + \sqrt{b})^2 ) simplifies exactly to ( a^2 + 2a\sqrt{