Verified answer

Step-by-step explanation:

To find rational numbers a and b in the expression you provided, let's simplify the expression step by step:

2 + √√32 - √√3 √√3 - 1_a + b√3 2 - √√3 2 + √√3 √√3 + 1

First, simplify the terms involving square roots:

√√32 is equivalent to 2√2, and √√3 is equivalent to √√(3), which simplifies to √√(3) = √(√3).

Now the expression looks like this:

2 + 2√2 - (√3 * √(√3)) - 1_a + b√3 2 - (√(√3)) 2 + (√3 * √(√3)) + 1

Now, simplify further:

2 + 2√2 - (√(3 * √3)) - 1_a + b√3 2 - (√(√3)) 2 + (√(3 * √3)) + 1

Simplify the expressions inside the square roots:

2 + 2√2 - (√(3√3)) - 1_a + b√3 2 - (√(√3)) 2 + (√(3√3)) + 1

Now, let's group the like terms:

(2 - 1) + (2√2 - √(3√3)) - (√(√3) + √(3√3)) + (b√3 + 1)

Combine the constants:

1 + (2√2 - √(3√3)) - (√(√3) + √(3√3)) + (b√3 + 1)

Now, we want this expression to be in the form a + b√3, where a and b are rational numbers. To achieve this, we need to simplify further:

(1 + 2√2 - √(3√3)) - (√(√3) + √(3√3)) + (b√3 + 1)

Now, let's collect the terms with √3 and those without √3 separately:

(2√2 - √(3√3) - √(√3) - √(3√3)) + (1 + 1 + b√3)

Combine like terms:

(2√2 - √(3√3) - √(√3) - √(3√3)) + (2 + b√3)

Now, we want the expression to be in the form a + b√3, so we equate the terms with and without √3 separately:

2√2 - √(3√3) - √(√3) - √(3√3) = b√3

2 + b√3 = a

Now, you can compare the coefficients:

For the term with √3:

b = -1

For the term without √3:

a = 2

So, the rational numbers a and b are:

a = 2

b = -1