Question

please answer this also

Answer 1

Answer :

x⁴ + 4/x⁴ = 12

Solution :

• Given : x = √(2 + √2)
• To find : x⁴ + 4/x⁴ = ?

We have ;

x = √(2 + √2)

Squaring both the sides , we get ;

=> x² = [ √(2 + √2) ]²

=> x² = 2 + √2

=> x² - 2 = √2

Again ,

Squaring both the sides , we get ;

=> (x² - 2)² = (√2)²

=> (x²)² - 2•x²•2 + 2² = 2

=> x⁴ - 4x² + 4 = 2

=> x⁴ + 4 - 2 = 4x²

=> x⁴ + 2 = 4x²

Now ,

Dividing both the sides by x² , we get ;

=> (x⁴ + 2) / x² = 4x² / x²

=> x⁴/x² + 2/x² = 4

=> x² + 2/x² = 4

Now ,

Squaring both the sides , we get ;

=> (x² + 2/x²)² = 4²

=> (x²)² + 2•x²•(2/x²) + (2/x²)² = 4²

=> x⁴ + 4 + 4/x² = 16

=> x⁴ + 4/x⁴ = 16 - 4

=> x⁴ + 4/x⁴ = 12

Hence ,

x⁴ + 4/x⁴ = 12

Answer 2

Answer :

• Option 4 is correct answer , i.e 12 is correct answer

Solution :

Method : 1

• Given , x = √(2+✓2)

Squaring on both sides

• x² = (√(2+√2))²
• x² = 2+√2-----(1)

Again squaring on both sides ,

• x⁴ = (2+√2)²

Using the formula (a+b)²= a²+b²+2ab ,

• x⁴ = 2² + (✓2)² + 2(2)√2
• x⁴ = 4 + 2 + 4√2
• x⁴ = 6 + 4√2

Required to find :

• x⁴ + 4/x⁴ = ?

=> x⁴ + 4/x⁴ = (6+4√2) + 4/(6+4√2)

=> x⁴ + 4/x⁴ = {(6+4√2)²+4}/ (6+4√2)

Using the formula (a+b)²= a²+b²+2ab ,

=> x⁴ + 4/x⁴ = {36+32+48√2+4}/(6+4√2)

=> x⁴ + 4/x⁴ = {72+48√2}/(6+4√2)

=> x⁴ + 4/x⁴ = 2(36+24√2) / 2(3+2√2)

Canceling 2 both on numerator and denominator

=> x⁴ + 4/x⁴ = (36+24√2) / (3+2√2)

Rationalising the denominator ,

Multiplying and dividing by (3-2√2)

=> x⁴ + 4/x⁴ = (36+24√2) / (3+2√2) * (3-2√2)/(3-2√2)

=> x⁴ + 4/x⁴ = (36+24√2) (3-2√2) / (3+2√2) (3-2√2)

Using the formula (a+b)(a-b) = a² - b² in the denominator,

=> x⁴ + 4/x⁴ = (36+24√2) (3-2√2) / 3² - (2√2)²

=> x⁴ + 4/x⁴ = (36+24√2) (3-2√2) / 9-8

=> x⁴ + 4/x⁴ = (36+24√2) (3-2√2) / 1

=> x⁴ + 4/x⁴ = (36+24√2) (3-2√2)

=> x⁴ + 4/x⁴ = 36(3) + 24√2(3) - 36(2√2) - 24√2(2√2)

=> x⁴ + 4/x⁴ = 108 + 72√2 -72√2 - 96

=> x⁴ + 4/x⁴ = 108 - 96

=> x⁴ + 4/x⁴ = 12

Therefore the value of x⁴ + 4/x⁴ = 12