Question

If x = (4096)^7+4 √3, then which of the following equals 64?
1. \(\frac{{{x^{7/2}}}}{{{x^{2\sqrt 3 }}}}\)
2. \(\frac{{{x^{\frac{7}{2}}}}}{{{x^{\frac{4}{{\sqrt 3 }}}}}}\)
3. \(\frac{{{x^{\frac{7}{2}}}}}{{{x^{\frac{4}{{2\sqrt 3 }}}}}}\)
4. \(\frac{{{x^7}}}{{{x^{4\sqrt 3 }}}}\)

Answer 1

Correct Answer - Option 1 : \(\frac{{{x^{7/2}}}}{{{x^{2\sqrt 3 }}}}\)

Calculation:

Let us solve step by step

 x = (4096)^{7 + 4 √3}

⇒ x^{1/(7 + 4√3)} = 64²   

When we rationalize 1/(7 + 4√3) we get 7 - 4√3

⇒ x^{(7 - 4√3)/2} = 64

⇒ 64 = \(\frac{{{x^{7/2}}}}{{{x^{2\sqrt 3 }}}}\)

∴ The correct answer is \(\frac{{{x^{7/2}}}}{{{x^{2\sqrt 3 }}}}\).