Question

If \[\sqrt 2 = 1.414\] and \[\sqrt 5 = 2.236\], find the value of \[\dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }}\] up to three places of decimals.

Answer 1

Hint:
Here, we have to find the value of the given mathematical expression. We will simplify the given mathematical expression by rationalizing the denominator. Then we will substitute the values in the expression and solve it further to find the required value.

Formula Used:
Rule of surds: \[\sqrt {a \times b} = \sqrt a \times \sqrt b \]

Complete step by step solution:
We will find the value of \[\dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }}\]
Now, we will rationalize the denominator by multiplying the numerator and the denominator by \[\sqrt 2 \], we get
\[\dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}\]
By multiplying the terms, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{\sqrt 2 \left( {\sqrt {10} } \right) - \sqrt 2 \left( {\sqrt 5 } \right)}}{{2\sqrt 2 \times \sqrt 2 }}\]
Multiplying the surds, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{\left( {\sqrt {2 \times 10} } \right) - \left( {\sqrt {2 \times 5} } \right)}}{{2{{\left( {\sqrt 2 } \right)}^2}}}\]
By simplification, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{\left( {\sqrt {2 \times 2 \times 5} } \right) - \left( {\sqrt {2 \times 5} } \right)}}{{2 \times 2}}\]
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{2\left( {\sqrt 5 } \right) - \left( {\sqrt {2 \times 5} } \right)}}{4}\]

Now, using the rule of surds \[\sqrt {a \times b} = \sqrt a \times \sqrt b \], we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{2\left( {\sqrt 5 } \right) - \sqrt 2 \times \sqrt 5 }}{4}\]
Substituting \[\sqrt 2 = 1.414\] and \[\sqrt 5 = 2.236\], we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{2\left( {2.236} \right) - \left( {1.414} \right) \times \left( {2.236} \right)}}{4}\]
By multiplying the terms and simplifying the expression, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{4.472 - 3.1617}}{4}\]
Subtracting the terms, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{1.3103}}{4}\]
By dividing the numerator by denominator, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = 0.327575\]
Now, we will correct the decimal up to three decimals.
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} \simeq 0.327\]

Therefore, the value of \[\dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }}\] is \[0.327\]

Note:
We know that rationalizing the denominator is a method of eliminating the radical expressions in the denominator such as the square roots or cube roots by multiplying with the conjugate to make the denominator a rational number. We can also find the value by directly substituting the given values.
\[\dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{\sqrt {2 \times 5} - \sqrt 5 }}{{2\sqrt 2 }}\]
By using the rule of surds \[\sqrt {a \times b} = \sqrt a \times \sqrt b \], we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{\sqrt 2 \left( {\sqrt 5 } \right) - \sqrt 5 }}{{2\sqrt 2 }}\]
Substituting \[\sqrt 2 = 1.414\] and \[\sqrt 5 = 2.236\], we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{1.414\left( {2.236} \right) - \left( {2.236} \right)}}{{2 \times \left( {1.414} \right)}}\]
By multiplying the decimals, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{3.161704 - 2.236}}{{2.828}}\]
By subtracting the terms, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = \dfrac{{0.925704}}{{2.828}}\]
By dividing the terms, we get
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} = 0.32733\]
Now, we will correct the decimal up to three decimals.
\[ \Rightarrow \dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }} \simeq 0.327\]
Therefore, the value of \[\dfrac{{\sqrt {10} - \sqrt 5 }}{{2\sqrt 2 }}\] is \[0.327\].