Question

(1/b-√3 - b^2+6/b^3 - 3√3) : 3b/b^2+b√3+3

Answer 1

(b-3)(b²+3b+9)+(3b³+6b³):(-3b³)=b³-27-1-2=b³-30

Answer 2

To simplify the expression, let's follow the order of operations and combine like terms step by step.

Given expression:

(1/(b - √3) - (b^2 + 6)/(b^3 - 3√3)) : (3b/(b^2 + b√3 + 3))

Step 1: Simplify the numerator of the first fraction.

1.1: Find a common denominator for the first fraction's numerator. The common denominator for (b - √3) and (b^3 - 3√3) is (b^3 - √3).

1.2: Rewrite the first fraction's numerator with the common denominator.

Numerator of the first fraction = [(1 * (b^3 - √3))/(b^3 - √3)] - [(b^2 + 6)(b - √3)/(b^3 - √3)]

Step 2: Simplify the numerator of the second fraction.

Numerator of the second fraction = 3b

Step 3: Simplify the denominator of the entire expression.

Denominator = (b^2 + b√3 + 3)

Now, the expression becomes:

[(b^3 - √3) - (b^2 + 6)(b - √3)] / (b^3 - √3) : 3b / (b^2 + b√3 + 3)

Step 4: Distribute the terms in the numerator of the first fraction.

[(b^3 - √3) - (b^3 - √3 - 6b + 6√3)] / (b^3 - √3) : 3b / (b^2 + b√3 + 3)

Step 5: Simplify the numerator.

The (b^3 - √3) terms cancel out.

[-6b + 6√3] / (b^3 - √3) : 3b / (b^2 + b√3 + 3)

Step 6: Now, we need to divide the entire expression by (3b / (b^2 + b√3 + 3)).

Dividing by a fraction is the same as multiplying by its reciprocal.

So, the expression becomes:

[-6b + 6√3] / (b^3 - √3) * [(b^2 + b√3 + 3) / 3b]

Step 7: Factor out a 3 from the numerator.

3(-2b + 2√3) / (b^3 - √3) * [(b^2 + b√3 + 3) / 3b]

Step 8: Simplify the expression by canceling out common factors.

(-2b + 2√3) / (b^3 - √3) * (b^2 + b√3 + 3) / b

Step 9: Now, you can cancel out the common factor of "b" from the numerator and denominator.

(-2 + 2√3) / (b^2 - √3) * (b + √3)

And that's the simplified form of the given expression.