Question

√7-√3/√7+√3+√7+√3/√7-√3

Answer 1

Ответ:

$2 \sqrt{7}$

Answer 2

To simplify the given expression, let's rationalize the denominators first.

Step 1: Rationalize the denominator of the first fraction (√7 - √3) / (√7 + √3) To rationalize the denominator, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is (√7 - √3):

[(√7 - √3) / (√7 + √3)] * [(√7 - √3) / (√7 - √3)] = (√7 - √3)(√7 - √3) / [(√7)^2 - (√3)^2] = (√49 - 2√21 + 3) / (7 - 3) = (7 - 2√21 + 3) / 4 = (10 - 2√21) / 4 = 5 - √21/2

Step 2: Rationalize the denominator of the second fraction (√7 + √3) / (√7 - √3) To rationalize the denominator, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is (√7 + √3):

[(√7 + √3) / (√7 - √3)] * [(√7 + √3) / (√7 + √3)] = (√7 + √3)(√7 + √3) / [(√7)^2 - (√3)^2] = (√49 + 2√21 + 3) / (7 - 3) = (7 + 2√21 + 3) / 4 = (10 + 2√21) / 4 = 5 + √21/2

Now the expression becomes:

(5 - √21/2 + √7 + √3) / (5 + √21/2 + √7 + √3)

We can combine the terms with the same radicals:

= [(5 + √7) + (√21 - √21)/2] / [(5 + √7) + (√21 + √21)/2]

Since (√21 - √21) is equal to 0, it simplifies further:

= (5 + √7) / (5 + √7)

Finally, since the numerator and denominator are the same, the value of the expression is 1:

= 1