Question
If \(x = {(4096)^{7 + 4\sqrt 3 }}\), then which of the following equals 64 ?
- \(\frac{{{x^7}}}{{{x^{2\sqrt 3 }}}}\)
- \(\frac{{{x^7}}}{{{x^{4\sqrt 3 }}}}\)
- \(\frac{{{x^{\frac{7}{2}}}}}{{{x^{\frac{4}{{\sqrt 3 }}}}}}\)
- \(\frac{{{x^{\frac{7}{2}}}}}{{{x^{2\sqrt 3 }}}}\)
Option: 4
Solution:
Solution:
\(x = {(4096)^{7 + 4\sqrt 3 }}\)
\({x^{\frac{1}{{7 + 4\sqrt 3 }}}} = (4096)\)
On rationalizing \(7 + 4\sqrt 3 \), we get \(\frac{1}{{7 + 4\sqrt 3 }} = 7 - 4\sqrt 3 \)
\(\therefore {x^{7 - 4\sqrt 3 }} = {(64)^2}\)
\(\therefore 64 = {x^{\frac{{7 - 4\sqrt 3 }}{2}}}\)
\(64 = \frac{{{x^{\frac{7}{2}}}}}{{{x^{2\sqrt 3 }}}}\)
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