Question

(√5-√3)/(√5+√3)+(√5+√3)/(√5-√3)терміново!

Answer 1

$\frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } + \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = \frac{ 2(\sqrt{5} + \sqrt{3} )(\sqrt{5} - \sqrt{3} )}{(\sqrt{5} + \sqrt{3} )(\sqrt{5} - \sqrt{3} )} = 2$

Answer 2

To simplify the given expression, we can use a technique called "rationalizing the denominator." Rationalizing the denominator means eliminating any square roots in the denominator by multiplying the numerator and denominator by a suitable expression.

Let's simplify the expression step by step:

Expression: (√5 - √3)/(√5 + √3) + (√5 + √3)/(√5 - √3)

Step 1: Rationalize the denominators

To rationalize the first fraction's denominator (√5 + √3), we multiply the numerator and denominator by its conjugate (√5 - √3):

(√5 - √3) * (√5 - √3) = (√5)^2 - (√3)^2 = 5 - 3 = 2

Now, the first fraction becomes: (√5 - √3) * (√5 - √3) / 2

To rationalize the second fraction's denominator (√5 - √3), we multiply the numerator and denominator by its conjugate (√5 + √3):

(√5 + √3) * (√5 + √3) = (√5)^2 + (√3)^2 = 5 + 3 = 8

Now, the second fraction becomes: (√5 + √3) * (√5 + √3) / 8

Step 2: Combine the fractions

Now, the expression becomes:

(√5 - √3) * (√5 - √3) / 2 + (√5 + √3) * (√5 + √3) / 8

Step 3: Expand the expressions

(√5 - √3) * (√5 - √3) = (√5)^2 - 2√5√3 + (√3)^2 = 5 - 2√15 + 3 = 8 - 2√15

(√5 + √3) * (√5 + √3) = (√5)^2 + 2√5√3 + (√3)^2 = 5 + 2√15 + 3 = 8 + 2√15

Step 4: Substitute back into the expression

(8 - 2√15) / 2 + (8 + 2√15) / 8

Step 5: Simplify further

(8 - 2√15) / 2 can be simplified to 4 - √15

(8 + 2√15) / 8 can be simplified to 1 + (1/4)√15

Now, the final expression is:

4 - √15 + 1 + (1/4)√15

Step 6: Combine like terms

4 + 1 = 5

So, the simplified expression is:

5 + (1/4)√15

There is no way to further simplify this expression because the terms are unlike and cannot be combined. Hence, the simplified expression is:

5 + (1/4)√15