Question

If \[x = \sqrt 7 + \sqrt 3 \] and \[xy = 4\] , then \[{x^4} + {y^4}\]
A. 400
B. 368
C. 200
D. none of these

Answer 1

Hint: First express y in terms of x as \[y = \dfrac{4}{x}\]. Then find the value of y by substituting x as \[\sqrt 7 + \sqrt 3 \]. Substitute the values of x as \[\sqrt 7 + \sqrt 3 \]and y as \[\sqrt 7 - \sqrt 3 \] in formula \[{\left( {{x^2} + {y^2}} \right)^2} - 2{(xy)^2}\]and obtain the required value.

Formula used:
1. \[{a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab\]
2. \[(a+b)(a-b) = a^2 - b^2\]

Complete step by step solution:
It is given that \[x = \sqrt 7 + \sqrt 3 \].
Now,
\[\begin{array}{c}xy = 4\\y = \dfrac{4}{x}\end{array}\]
Substitute the values of x as \[\sqrt 7 + \sqrt 3 \],

\[\dfrac{4}{{\sqrt 7 + \sqrt 3 }} = \dfrac{{4\left( {\sqrt 7 - \sqrt 3 } \right)}}{{\left( {\sqrt 7 + \sqrt 3 } \right)\left( {\sqrt 7 - \sqrt 3 } \right)}}\]
\[ = \dfrac{{4\left( {\sqrt 7 - \sqrt 3 } \right)}}{{{{\left( {\sqrt 7 } \right)}^2} - {{\left( {\sqrt 3 } \right)}^2}}}\]
\[ = \dfrac{{4\left( {\sqrt 7 - \sqrt 3 } \right)}}{{7 - 3}}\]
\[ = \dfrac{{4\left( {\sqrt 7 - \sqrt 3 } \right)}}{4}\]
\[ = \sqrt 7 - \sqrt 3 \]
The formula of \[{a^2} + {b^2}\] is \[{\left( {a + b} \right)^2} - 2ab\].
Substitute \[{x^2}\] for a and \[{y^2}\]for b in the formula \[{a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab\] and calculate to obtain the required value.
Therefore,
\[{\left( {{x^2}} \right)^2} + {\left( {{y^2}} \right)^2} = {\left( {{x^2} + {y^2}} \right)^2} - 2{x^2}{y^2}\]
\[{x^4} + {y^4} = {\left( {{x^2} + {y^2}} \right)^2} - 2{x^2}{y^2}\]
\[ = {\left( {{x^2} + {y^2}} \right)^2} - 2{(xy)^2} - - - - - (1)\]
Now,
\[\begin{array}{l}x = \sqrt 7 + \sqrt 3 \\\therefore {x^2} = {\left( {\sqrt 7 + \sqrt 3 } \right)^2}\\ = {\left( {\sqrt 7 } \right)^2} + {\left( {\sqrt 3 } \right)^2} + 2.\sqrt 7 .\sqrt 3 \\ = 10 + 2\sqrt {21} \end{array}\]
And
\[\begin{array}{l}y = \sqrt 7 - \sqrt 3 \\\therefore {y^2} = {\left( {\sqrt 7 - \sqrt 3 } \right)^2}\\ = {\left( {\sqrt 7 } \right)^2} + {\left( {\sqrt 3 } \right)^2} - 2.\sqrt 7 .\sqrt 3 \\ = 10 - 2\sqrt {21} \end{array}\]
And
\[xy = (\sqrt 7 + \sqrt 3 )(\sqrt 7 - \sqrt 3 )\]
\[xy = {\left( {\sqrt 7 } \right)^2} - {\left( {\sqrt 3 } \right)^2}\]
\[xy = 7 - 3\\ = 4\]
From (1) we have,
\[{x^4} + {y^4} = {\left( {{x^2} + {y^2}} \right)^2} - 2{(xy)^2}\]
\[{x^4} + {y^4} = {\left[ {10 + 2\sqrt {21} + 10 - 2\sqrt {21} } \right]^2} - 2{(4)^2}\]
\[{x^4} + {y^4} = 400 - 32 = 368\]

The correct option is B.

Note: Students sometime get confused with these formulas of indices
\[{\left( {{x^m}} \right)^n} = {x^{mn}}{\rm{ and }}{x^m} + {x^n} = {x^{m + n}}\] , here we are using the formula \[{\left( {{x^m}} \right)^n} = {x^{mn}}\] .