Question

if x= 5-2√6,find the value of x^4 +1/x^4​

Answer 1

Answer:

the answer is 1.

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Answer 2

Given :-

x = 5-2√6

To find :-

The value of x⁴+(1/x⁴)

Solution :-

Given that x = 5-2√6 ----------(1)

=> 1/x = 1/(5-2√6)

The denominator = 5-2√6

We know that

The Rationalising factor of a-√b is a+√b

The Rationalising factor of 5-2√6 = 5+2√6

On Rationalising the denominator then

=> 1/x = [1/(5-2√6)]×[(5+2√6)/(5+2√6)]

=> 1/x = (5+2√6)/[(5-2√6)(5+2√6)]

=> 1/x = (5+2√6)/[5²-(2√6)²]

Since, (a+b)(a-b) = a²-b2

Where, a = 5 and b = 2√6

=> 1/x = (5+2√6)/(25-24)

=> 1/x = (5+2√6)/1

Therefore, 1/x = 5+2√6 ------------(2)

Now,

We know that

(a+b)² = a²+b²+2ab

=> a²+b² = (a+b)² - 2ab

Therefore,

x²+(1/x²) = [x+(1/x)]² - 2(x)(1/x)

=> x²+(1/x²) = [x+(1/x)]² - 2(x/x)

=> x²+(1/x²) = [x+(1/x)]² - 2(1)

=> x²+(1/x²) = [x+(1/x)]² - 2

=> x²+(1/x²) = (5-2√6+5+2√6)²-2

=> x²+(1/x²) = (5+5)²-2

=> x²+(1/x²) = (10)²-2

=> x²+(1/x²) = 100-2

=> x²+(1/x²) = 98 ----------------(3)

On squaring both sides then

=> [x²+(1/x²)]² = (98)²

=> (x²)²+(1/x²)²+2(x²)(1/x²) = 9604

Since, (a+b)² = a²+b²+2ab

Where, a = x² and b = 1/x²

=> x⁴+(1/x⁴)+2(x²/x²) = 9604

=> x⁴+(1/x⁴)+2 = 9604

=> x⁴+(1/x⁴) = 9604-2

=> x⁴+(1/x⁴) = 9602

Answer :-

The value of x⁴+(1/x⁴) = 9602

Used formulae:-

• (a+b)² = a²+2ab+b²
• (a+b)+a-b) = a²-b²
• The Rationalising factor of a-√b is a+√b