((frac (a-b)/(sqrt (a)+sqrt {b)))^3+2asqrt (a)+bsqrt (b)}(3a^2+3bsqrt (ab))+(sqrt (ab)-a)/(asqrt (a)-bsqrt (a))

Question

((frac (a-b)/(sqrt (a)+sqrt {b)))^3+2asqrt (a)+bsqrt (b)}(3a^2+3bsqrt (ab))+(sqrt (ab)-a)/(asqrt (a)-bsqrt (a))

Answer 1

We are tasked with simplifying the given expression:

### Step 1: Simplify the first termThe numerator of the first fraction is:

Let us simplify

. Multiply numerator and denominator by

(the conjugate of the denominator):

Since

, we cancel

in the numerator and denominator:

Thus, the cube of this expression is:

Using the binomial expansion for \((x-y)^3\), we have:

which simplifies to:

Now, the numerator of the first term becomes:

Combine like terms:

which simplifies to:

The denominator of the first term is:

Factor out

from both terms:

Thus, the first term becomes:

Cancel the common factor of

in the numerator and denominator:

### Step 2: Simplify the second termThe second term is:

Factor out

from the denominator:

Thus, the second term becomes:

Factor out

from the numerator:

So the second term becomes:

Cancel

in the numerator and denominator:

Factor

from the numerator:

Multiply numerator and denominator by

:

### Step 3: Combine the two termsThe full expression is now:

Notice that the second term has a denominator of

, which is the negative of

. Rewrite the second term as:

Now combine the two terms over a common denominator. The least common denominator (LCD) is \((a^2 + b\sqrt{ab})(a-b)\). Rewrite each fraction with this denominator.For the first term:

For the second term:

Combine the two fractions:

Simplify the numerator (this involves expanding and combining like terms). After simplification, the entire expression reduces to:

### Final Answer:

Question

((frac (a-b)/(sqrt (a)+sqrt {b)))^3+2asqrt (a)+bsqrt (b)}(3a^2+3bsqrt (ab))+(sqrt (ab)-a)/(asqrt (a)-bsqrt (a))

Solution

Resposta