Question
Simplify $\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$
✓
Answer
$\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$
$=\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}\times\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}\times\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}}$
$=\frac{\big(\sqrt{13}-\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}+\frac{\big(\sqrt{13}+\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}$
$=\frac{13+11-2\times\sqrt{13}\times\sqrt{11}}{13-11}+\frac{13+11+2\times\sqrt{13}\times\sqrt{11}}{13-11}$
$=\frac{24-2\sqrt{143}}{2}+\frac{24+2\sqrt{143}}{2}$
$=\frac{24-2\sqrt{143}+24+2\sqrt{143}}{2}$
$=\frac{48}{2}$ $=24$
$=\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}\times\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}\times\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}}$
$=\frac{\big(\sqrt{13}-\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}+\frac{\big(\sqrt{13}+\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}$
$=\frac{13+11-2\times\sqrt{13}\times\sqrt{11}}{13-11}+\frac{13+11+2\times\sqrt{13}\times\sqrt{11}}{13-11}$
$=\frac{24-2\sqrt{143}}{2}+\frac{24+2\sqrt{143}}{2}$
$=\frac{24-2\sqrt{143}+24+2\sqrt{143}}{2}$
$=\frac{48}{2}$ $=24$
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
Prove that the angle in a segment greater than a semi-circle is less than a right angle.
Over the past $200$ working days, the number of defective parts produced by a machine is given in the following table:
Determine the probability that tomorrow’s output will have:
$i.$ No defective part.
$ii.$ Atleast one defective part.
$iii.$ Not more than $5$ defective parts.
$iv.$ More than $13$ defective parts.
→| Number of defective parts | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ |
| Days | $50$ | $32$ | $22$ | $18$ | $12$ | $12$ | $10$ | $10$ | $10$ | $8$ | $6$ | $6$ | $2$ | $2$ |
$i.$ No defective part.
$ii.$ Atleast one defective part.
$iii.$ Not more than $5$ defective parts.
$iv.$ More than $13$ defective parts.
ABCD ia a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that:
- AD || BC.
- EB = EC.
Find the area of the shaded region in fig. below
Points $A(5, 3), B(-2, 3)$ and $D(5, -4)$ are three vertices of a square $ABCD.$ Plot these points on a graph paper and hence find the coordinates of the vertex $C.$
→The radii of two cylinders are in the ratio $2:3$ and their heights are in the ratio $5:3$. Calculate the ratio of their volumes and the ratio of their curved surfaces.
→ Rationalise the denominator of the following:
$\frac{4}{2+\sqrt{3}+\sqrt{7}}$
→→$\frac{4}{2+\sqrt{3}+\sqrt{7}}$
If $\text{p}=\frac{3-\sqrt{5}}{3+\sqrt{5}}$ and $\text{q}=\frac{3+\sqrt{5}}{3-\sqrt{5}},$ find the value of $p^2 + q^2.$
→Three girls Ishita, Isha and Nisha are playing a game by standing on a circle of radius $20\ m$ drawn in a park. Ishita throws a ball to Isha, Isha to Nisha and Nisha to Ishita. If the distance between Ishita and Isha and between Isha and Nisha is $24\ m$ each, what is the distance between Ishita and Nisha.
→