Question

Evaluate $ {\left( {\dfrac{a}{{2b}} + \dfrac{{2b}}{a}} \right)^2} - {\left( {\dfrac{a}{{2b}} - \dfrac{{2b}}{a}} \right)^2} - 4 $ .

Answer 1

Hint: Use the property of $ {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab $ and also $ {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab $ to simplify the given expression and then use the calculations to simplify further the expression.

Complete step-by-step answer:
 Use the properties of $ {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab $ and also $ {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab $ to simplify the given expression.
The value of the $ {\left( {\dfrac{a}{{2b}} + \dfrac{{2b}}{a}} \right)^2} $ is equal to
$ {\left( {\dfrac{a}{{2b}}} \right)^2} + {\left( {\dfrac{{2b}}{a}} \right)^2} + 2 \times \dfrac{a}{{2b}} \times \dfrac{{2b}}{a} $ by the above mentioned property and the value of the $ {\left( {\dfrac{a}{{2b}} - \dfrac{{2b}}{a}} \right)^2} $ is equal to $ {\left( {\dfrac{a}{{2b}}} \right)^2} + {\left( {\dfrac{{2b}}{a}} \right)^2} - 2 \times \dfrac{a}{{2b}} \times \dfrac{{2b}}{a} $ . Substitute these values in the given expression and simplify.
 $
  {\left( {\dfrac{a}{{2b}} + \dfrac{{2b}}{a}} \right)^2} - {\left( {\dfrac{a}{{2b}} - \dfrac{{2b}}{a}} \right)^2} - 4 = {\left( {\dfrac{a}{{2b}}} \right)^2} + {\left( {\dfrac{{2b}}{a}} \right)^2} + 2 \times \dfrac{a}{{2b}} \times \dfrac{{2b}}{a} - \left( {{{\left( {\dfrac{a}{{2b}}} \right)}^2} + {{\left( {\dfrac{{2b}}{a}} \right)}^2} - 2 \times \dfrac{a}{{2b}} \times \dfrac{{2b}}{a}} \right) - 4 \\
   = {\left( {\dfrac{a}{{2b}}} \right)^2} + {\left( {\dfrac{{2b}}{a}} \right)^2} + 2 - {\left( {\dfrac{a}{{2b}}} \right)^2} - {\left( {\dfrac{{2b}}{a}} \right)^2} + 2 - 4 \\
   = 2 + 2 - 4 \\
   = 0 \;
  $
Hence, the value of the expression $ {\left( {\dfrac{a}{{2b}} + \dfrac{{2b}}{a}} \right)^2} - {\left( {\dfrac{a}{{2b}} - \dfrac{{2b}}{a}} \right)^2} - 4 $ is equal to $ 0 $ .
Also remember that the value of the expression $ {\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} $ is always equal to the value $ 4ab $ .
So, verify the value of the expression $ {\left( {\dfrac{a}{{2b}} + \dfrac{{2b}}{a}} \right)^2} - {\left( {\dfrac{a}{{2b}} - \dfrac{{2b}}{a}} \right)^2} - 4 $ by using this property:
 $
  {\left( {\dfrac{a}{{2b}} + \dfrac{{2b}}{a}} \right)^2} - {\left( {\dfrac{a}{{2b}} - \dfrac{{2b}}{a}} \right)^2} - 4 = 4 \times \dfrac{a}{{2b}} \times \dfrac{{2b}}{a} - 4 \\
   = 4 \times 1 - 4 \\
   = 4 - 4 \\
   = 0 \;
  $
It also yields the same value for the expression given in the question.
So, the correct answer is “0”.

Note: Also remember that the value of the expression $ {\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} $ is always equal to the value $ 4ab $ . Use this property and verify the result it will give the same result as above. Using identity is always preferable since it reduces the complexity of simplification.