Question

Please answer yourself
5+√2/5-√2 + 5-√2/5+√2 simplify?

Answer 1

Solution!!

[(5 + √2)/(5 - √2)] + [(5 - √2)/(5 + √2)]

Rationalise the denominator of the first fraction.

= [(5 + √2)/(5 - √2) × (5+ √2)/(5 + √2)] + [(5 - √2)/(5 + √2)]

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

= [(5 + √2)²/(5² - √2²)] + [(5 - √2)/(5 + √2)]

= [(25 + 2 + 10√2)/(25 - 2)] + [(5 - √2)/(5 + √2)]

= [(27 + 10√2)/23] + [(5 - √2)/(5 + √2)]

Rationalise the denominator of the second fraction.

= [(27 + 10√2)/23] + [(5 - √2)/(5 + √2) × (5 - √2)/(5 - √2)]

= [(27 + 10√2)/23] + [(5 - √2)(5 - √2)/(5 + √2)(5 - √2)]

= [(27 + 10√2)/23] + [(5 - √2)²/(5² - 2)]

= [(27 + 10√2)/23] + [(25 + 2 - 10√2)/(25 - 2)]

= [(27 + 10√2)/23] + [(27 - 10√2)/23]

As the denominators are same, we can write all the numerators above a common denominator.

= [(27 + 10√2) + (27 - 10√2)]/23

= [27 + 10√2 + 27 - 10√2]/23

= [54]/23

= 54/23

Identities used:-

(a + b)(a - b) = a² - b²

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab