Question

If x=3√3 + √26, find the value of 1/2(x+1/x).

$x = 3 \sqrt{3} + \sqrt{26}$
$find \: value \: of \: \frac{1}{2} (x + \frac{1}{x} )$
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Answer 1

Question: If x = 3√3 + √26, find the value of 1/2(x + 1/x).

Answer:

3√3

Step-by-step explanation:

In this problem we have the value of x, i.e. (3√3 + 26) and are asked to find the value of 1/2(x + 1/x). We can simply substitute the value of x into 1/2(x + 1/x), but it is a bit tricky. Because we cannot easily multiply and add the square root of the denominator. To make it easier, first rationalize the denominator and then add the resulting value to get the final answer.

Rationalise means removal of square root from denominator. To do so, multiply both numerator and denominator with same number. Example: Rationalisation of 1/√3 is √3/3 (it can be written as 1/√3 ×√3/√3 = √3/(√3)² = √3/3. Rationalisation of 3/(2 + √3) is 3(2 - √3) or 6 - 3√3 [Conjugate of (2 + √3) is (2 - √3); sign in between the expression changes].

Back to the question!

→ x = 3√3 + √26

→ 1/x = 1/(3√3 + √26)

Now, rationalise the denominator. From above we know that conjugate of (3√3 + √26) is (3√3 - √26). So,

→ 1/x = 1/(3√3 + √26) × (3√3 - √26)/(3√3 - √26)

→ 1/x = (3√3 - √26)/[(3√3 + √26)(3√3 - √26)

→ 1/x = (3√3 - √26)/[(3√3)² - (√26)²]

Used identity: (a + b)(a - b) = a² - b²

→ 1/x = (3√3 - √26)/(9*3 - 26)

→ 1/x = (3√3 - √26)/(27 - 26)

→ 1/x = (3√3 - √26)/1

→ 1/x = 3√3 - √26

Now, we have the value of both x and 1/x. Simply, substitute the value of x and 1/x in 1/2(x + 1/x).

→ 1/2(x + 1/x) = 1/2 × (3√3 + √26 + 3√3 - √26)

+√26 - √26 cancel out, we left with

→ 1/2(x + 1/x) = 1/2 × (3√3 + 3√3)

→ 1/2(x + 1/x) = 1/2 × (6√3)

→ 1/2(x + 1/x) = 3√3

Hence, the value of 1/2(x + 1/x) is 3√3.

Answer 2

Answer:

3√3

Step-by-step explanation:

see the picture for explaination, hope it helps