Question
Rationalize the denominator of $\frac{1}{{\sqrt 7 - \sqrt 6 }}$
✓
Answer
$\frac{1}{{\sqrt 7 - \sqrt 6 }}$
We need to multiply the numerator and denominator of $\frac{1}{{\sqrt 7 - \sqrt 6 }}$ by $\sqrt 7 + \sqrt 6,to get \frac{1}{{\sqrt 7 - \sqrt 6 }} \times \frac{{\sqrt 7 + \sqrt 6 }}{{\sqrt 7 + \sqrt 6 }} = \frac{{\sqrt 7 + \sqrt 6 }}{{\left( {\sqrt 7 - \sqrt 6 } \right)\left( {\sqrt 7 + \sqrt 6 } \right)}}$
We need to apply the formula $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$ in the denominator to get
$\frac{1}{{\sqrt 7 - \sqrt 6 }} = \frac{{\sqrt 7 + \sqrt 6 }}{{{{\left( {\sqrt 7 } \right)}^2} - {{\left( {\sqrt 6 } \right)}^2}}}$
$= \frac{{\sqrt 7 + \sqrt 6 }}{{7 - 6}}$
$= \sqrt 7 + \sqrt 6 .$
Therefore, we conclude that on rationalizing the denominator of $\frac{1}{{\sqrt 7 - \sqrt 6 }}$ we get $\sqrt 7 + \sqrt 6$ .
We need to multiply the numerator and denominator of $\frac{1}{{\sqrt 7 - \sqrt 6 }}$ by $\sqrt 7 + \sqrt 6,to get \frac{1}{{\sqrt 7 - \sqrt 6 }} \times \frac{{\sqrt 7 + \sqrt 6 }}{{\sqrt 7 + \sqrt 6 }} = \frac{{\sqrt 7 + \sqrt 6 }}{{\left( {\sqrt 7 - \sqrt 6 } \right)\left( {\sqrt 7 + \sqrt 6 } \right)}}$
We need to apply the formula $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$ in the denominator to get
$\frac{1}{{\sqrt 7 - \sqrt 6 }} = \frac{{\sqrt 7 + \sqrt 6 }}{{{{\left( {\sqrt 7 } \right)}^2} - {{\left( {\sqrt 6 } \right)}^2}}}$
$= \frac{{\sqrt 7 + \sqrt 6 }}{{7 - 6}}$
$= \sqrt 7 + \sqrt 6 .$
Therefore, we conclude that on rationalizing the denominator of $\frac{1}{{\sqrt 7 - \sqrt 6 }}$ we get $\sqrt 7 + \sqrt 6$ .
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Factorise:
$9 x^2+18 x+8$
→$9 x^2+18 x+8$
In how many lines two distinct planes can intersect?
How many triangles can be drawn having its angles as $53^{\circ}, 64^{\circ}$ and $63^{\circ}$ ? Give reason for your answer.
→Express the following rational numbers as decimals: $\frac{2}{3}$
→If $O$ is the centre of the circle, find the value of $x$ in given figure :
→A hollow spherical shell is made of a metal of density $4.5\ g\ per\ cm ^3$. If its internal and external radii are $8 \ cm$ and $9 \ cm$ respectively, find the weight of the shell.
→In figure, if $P Q \| S T, \angle P Q R=110^{\circ}$ and $\angle R S T=130^{\circ}$. Find $\angle Q R S$.
→Solve the following equations for $x:$
$2^{\text{x}+1}=4^{\text{x}-3}$
→$2^{\text{x}+1}=4^{\text{x}-3}$
Simplify the expression: $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$
→Simplify the following expressions:
$(x+y-2 z)^2-x^2-y^2-3 z^2+4 x y$
→$(x+y-2 z)^2-x^2-y^2-3 z^2+4 x y$