Question

How do you simplify \[\dfrac{7-\sqrt{7}}{10+\sqrt{3}}\]?

Answer 1

Hint: Assume the given expression as ‘E’. Now, rationalize the denominator of this expression by multiplying it with its conjugate \[10-\sqrt{3}\]. Now, to balance this expression multiply the numerator also with this conjugate value. Simplify the denominator using the algebraic identity: \[\left( a+b \right)\left( a-b \right)=\left( {{a}^{2}}-{{b}^{2}} \right)\] to get the answer.

Complete step by step solution:
Here, we have been provided with the expression \[\dfrac{7-\sqrt{7}}{10+\sqrt{3}}\] and we are asked to simplify it. Let us assume this expression as ‘E’, so we have,
\[\Rightarrow E=\dfrac{7-\sqrt{7}}{10+\sqrt{3}}\]
Now, to simplify the above expression means we have to rationalize the denominator. We apply the rationalization process for the fractions having an irrational number in the denominator. We multiply the denominator with its conjugate to make it a rational number, particularly an integer. However, we can leave the numerator part.
In the above expression we have the denominator as \[10+\sqrt{3}\] whose conjugate is given as \[10-\sqrt{3}\]. So, multiplying the denominator with \[10-\sqrt{3}\] and balancing the value of expression by multiplying the numerator also with the same conjugate, we get,
\[\Rightarrow E=\dfrac{7-\sqrt{7}}{10+\sqrt{3}}\times \dfrac{10-\sqrt{3}}{10-\sqrt{3}}\]
In the denominator the above expression is of the form \[\left( a+b \right)\times \left( a-b \right)\] whose simplified form is given by the algebraic identity: \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]. So, we have the expression as: -
\[\begin{align}
  & \Rightarrow E=\dfrac{\left( 7-\sqrt{7} \right)\left( 10-\sqrt{3} \right)}{{{10}^{2}}-{{\left( \sqrt{3} \right)}^{2}}} \\
 & \Rightarrow E=\dfrac{\left( 7-\sqrt{7} \right)\left( 10-\sqrt{3} \right)}{100-3} \\
 & \Rightarrow E=\left( \dfrac{70+\sqrt{21}-10\sqrt{7}-7\sqrt{3}}{97} \right) \\
\end{align}\]
Hence, the above expression is our answer.

Note: One may note that we can also rationalize the numerator \[\left( 7-\sqrt{7} \right)\] by multiplying and dividing the expression with its conjugate \[\left( 7+\sqrt{7} \right)\], but if we do so then we will again get an irrational number in the denominator, so it will be better not to do so. You must know the meaning of rationalization and conjugate of a number otherwise it will be difficult to simplify the expression. Remember the important algebraic identities so that we can simplify the denominator.