Question

1) ((m-(2*sqrt(mn))+n)/(m-n)) 2) ((a+3*sqrt(a))/(a-9) 3) a/(a-1)-(sqrt(a))/(sqrt(a)+1)

Answer 1

1)
$\frac{m - 2 \sqrt{mn} + n }{m - n} = \frac{ {( \sqrt{m} - \sqrt{n} )}^{2} }{( \sqrt{m} - \sqrt{n} )( \sqrt{m} + \sqrt{n}) } = \frac{ \sqrt{m} - \sqrt{n} }{ \sqrt{m} + \sqrt{n} }$
2)
$\frac{a + 3 \sqrt{a} }{a - 9} = \frac{ \sqrt{a}( \sqrt{a} + 3)}{( \sqrt{a} - 3)( \sqrt{a} + 3)} = \frac{ \sqrt{a} }{ \sqrt{a} - 3}$
3)
$\frac{a}{a - 1} - \frac{ \sqrt{a} }{ \sqrt{a} + 1 } = \frac{a}{( \sqrt{a} - 1)( \sqrt{a} + 1) } - \frac{ \sqrt{a}( \sqrt{a} - 1) }{( \sqrt{a} + 1)( \sqrt{a} - 1) } = \frac{a - a + \sqrt{a} }{a - 1} = \frac{ \sqrt{a} }{a - 1}$

Answer 2

To simplify each of the given expressions, I'll provide step-by-step solutions:

1. ((m - 2*sqrt(mn) + n) / (m - n))

Step 1: Factor out 'm - n' from the numerator. (m - 2sqrt(mn) + n) = (m - n - 2sqrt(mn))

Step 2: Now, the expression becomes: ((m - n - 2*sqrt(mn)) / (m - n))

Step 3: Cancel out the common term '(m - n)' from the numerator and denominator. The final simplified expression is: 1 - 2*sqrt(mn)

2. ((a + 3*sqrt(a)) / (a - 9))

This expression seems to be already simplified. There are no common factors to be canceled out.

The expression remains: ((a + 3*sqrt(a)) / (a - 9))

3. (a / (a - 1)) - (sqrt(a) / (sqrt(a) + 1))

Step 1: To combine the fractions, find a common denominator for both terms, which is (a - 1)(sqrt(a) + 1).

Step 2: Rewrite the fractions with the common denominator: (a*(sqrt(a) + 1) / ((a - 1)(sqrt(a) + 1))) - (sqrt(a)*(a - 1) / ((a - 1)(sqrt(a) + 1)))

Step 3: Combine the fractions: ((asqrt(a) + a) - (sqrt(a)(a - 1))) / ((a - 1)(sqrt(a) + 1))

Step 4: Distribute and simplify: (a*sqrt(a) + a - sqrt(a)*a + sqrt(a)) / ((a - 1)(sqrt(a) + 1))

Step 5: Combine like terms: (sqrt(a) + a) / ((a - 1)(sqrt(a) + 1))

Now, this expression is simplified. There are no common factors to be canceled out.