Question

Find the square root of the following surd: $ 8 - 3\sqrt 7 $ .

Answer 1

Hint: As we know that square root can be defined as a number which when multiplied by itself gives a number as the product. For example $ 5*5 = 25 $ , here square root of $ 25 $ is $ 5 $ . There is no such formula to calculate square root formula but two ways are generally considered. They are the prime factorization method and division method. The symbol $ \sqrt {} $ is used to denote square roots and this symbol of square roots is also known as radical.

Complete step by step solution:
Here we have to find the value of $ 8 - 3\sqrt 7 $ . First we will multiply the numerator and denominator with $ \sqrt 2 $ . So we can write it as
 $ \dfrac{{\sqrt {8 \times \sqrt 2 - 3\sqrt 7 \times \sqrt 2 } }}{{\sqrt 2 }} = \dfrac{1}{{\sqrt 2 }}\sqrt {16 - 6\sqrt 7 } $ ( By taking the denominator out) .
We know the algebraic difference formula i.e. $ {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab $ . So we write the numerator in terms of this formula, we can write the numerator as
 $ 16 - 6\sqrt 7 = {\left( {\sqrt 7 } \right)^2} + {3^2} - 2(3)(\sqrt 7 ) $ .
 So we have
 $ \dfrac{1}{{\sqrt 2 }}\sqrt {{{(\sqrt 7 )}^2} + {3^2} - 2(3)(\sqrt {7)} } $ .
It can be written as
 $ \pm \dfrac{1}{{\sqrt 2 }}\sqrt {{{\left( {\sqrt 7 - 3} \right)}^2}} $ .
We know that $ \sqrt {{a^2}} $ can be written as $ a $ , because of the exponential formula, as the value of $ \sqrt {} $ is $ \dfrac{1}{2} $ , so it turns into $ {a^{2 \times \dfrac{1}{2}}} = a $ .
Hence the required answer is $ \pm \dfrac{1}{{\sqrt 2 }}\left( {\sqrt 7 - 3} \right) $ .
So, the correct answer is “ $ \pm \dfrac{1}{{\sqrt 2 }}\left( {\sqrt 7 - 3} \right) $ ”.

Note: Before solving this kind of question we should be aware of the algebraic formula and exponential rules. The number written inside the square root symbol or radical is known as radicand. We know that all real numbers have two square roots, one is a positive square root and another one is a negative square root. The positive square root is also referred to as the principal square root.