Question

If x=7+4√3, then what is the value of (x^3/x^6 + 3x^3 + 1)
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Answer 1

Answer:

x = 7+4√3

To find √x we proceed,

√x = √(7+4√3)

√x = √(7+2x2√3)

√x = √(7+2√3x4)

√x = √(3+4+2√3x4)….. {writing 7 = 3+4}

If we observe RHS of √x we observe form of

√(a² + b² +2ab) where a=√3 and b =√4

Hence, √x =√(√3 +√4)² = √3 + √4 = 2+√3

√x = 2+√3

1/√x = 1/(2+√3)

Multiplying both numerator and denominator by 2 - √3, we get

1/√x = (2-√3)/(2-√3)(2+√3) = (2-√3)/(2²-√3²) =

1/√x = 2-√3

Hence √x +1/√x = 2+√3 +2 -√3 = 4