Question
Simplify:
$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$
$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$
✓
Answer
For rationalising the denominator of a number, we multilply its numerator and denominator by its rationalising factor.
If a and b are integers, then $\big(\text{a}+\sqrt{\text{b}}\big)$ and $\big(\text{a}-\sqrt{\text{b}}\big)$ are rationalising factor of each other, as $\big(\text{a}+\sqrt{\text{b}}\big)\big(\text{a}-\sqrt{\text{b}}\big)=\big(\text{a}^2-\text{b}\big),$ which is rational.
Let us rationalise the denominator of the first term on the left hand side.
We have,
$\frac{4+\sqrt{5}}{4-\sqrt{5}}=\frac{4+\sqrt{5}}{4-\sqrt{5}}\times\frac{4+\sqrt{5}}{4+\sqrt{5}}$
$=\frac{\big(4+\sqrt{5}\big)^2}{(4)^2-\big(\sqrt{5}\big)^2}$
$=\frac{(4)^2+2(4)\big(\sqrt{5}\big)+\big(\sqrt{5}\big)^2}{16-5}$
$\frac{4+\sqrt{5}}{4-\sqrt{5}}=\frac{16+8\sqrt{5}+5}{11}=\frac{21+8\sqrt{5}}{11} \ ...(1)$
Now consider the denominator of the second term on the left hand side:
$\frac{4-\sqrt{5}}{4+\sqrt{5}}=\frac{4-\sqrt{5}}{4+\sqrt{5}}\times\frac{4-\sqrt{5}}{4-\sqrt{5}}$
$=\frac{\big(4-\sqrt{5}\big)^2}{(4)^2-\big(\sqrt{5}\big)^2}$
$=\frac{(4)^2-2(4)\big(\sqrt{5}\big)+\big(\sqrt{5}\big)^2}{16-5}$
$\frac{4-\sqrt{5}}{4+\sqrt{5}}=\frac{16-8\sqrt{5}+5}{11}=\frac{21-8\sqrt{5}}{11} \ ...(2)$
Adding equations (1) and (2), we have,
$\therefore \ \frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}=\frac{21+8\sqrt{5}}{11}=\frac{21-8\sqrt{5}}{11}$
$=\frac{21+8\sqrt{5}+21-8\sqrt{5}}{11}=\frac{42}{11}$
If a and b are integers, then $\big(\text{a}+\sqrt{\text{b}}\big)$ and $\big(\text{a}-\sqrt{\text{b}}\big)$ are rationalising factor of each other, as $\big(\text{a}+\sqrt{\text{b}}\big)\big(\text{a}-\sqrt{\text{b}}\big)=\big(\text{a}^2-\text{b}\big),$ which is rational.
Let us rationalise the denominator of the first term on the left hand side.
We have,
$\frac{4+\sqrt{5}}{4-\sqrt{5}}=\frac{4+\sqrt{5}}{4-\sqrt{5}}\times\frac{4+\sqrt{5}}{4+\sqrt{5}}$
$=\frac{\big(4+\sqrt{5}\big)^2}{(4)^2-\big(\sqrt{5}\big)^2}$
$=\frac{(4)^2+2(4)\big(\sqrt{5}\big)+\big(\sqrt{5}\big)^2}{16-5}$
$\frac{4+\sqrt{5}}{4-\sqrt{5}}=\frac{16+8\sqrt{5}+5}{11}=\frac{21+8\sqrt{5}}{11} \ ...(1)$
Now consider the denominator of the second term on the left hand side:
$\frac{4-\sqrt{5}}{4+\sqrt{5}}=\frac{4-\sqrt{5}}{4+\sqrt{5}}\times\frac{4-\sqrt{5}}{4-\sqrt{5}}$
$=\frac{\big(4-\sqrt{5}\big)^2}{(4)^2-\big(\sqrt{5}\big)^2}$
$=\frac{(4)^2-2(4)\big(\sqrt{5}\big)+\big(\sqrt{5}\big)^2}{16-5}$
$\frac{4-\sqrt{5}}{4+\sqrt{5}}=\frac{16-8\sqrt{5}+5}{11}=\frac{21-8\sqrt{5}}{11} \ ...(2)$
Adding equations (1) and (2), we have,
$\therefore \ \frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}=\frac{21+8\sqrt{5}}{11}=\frac{21-8\sqrt{5}}{11}$
$=\frac{21+8\sqrt{5}+21-8\sqrt{5}}{11}=\frac{42}{11}$
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
It being given that $\sqrt{3}=1.732,\sqrt{5}=2.236,\sqrt{6}=2.449$ and $\sqrt{10}=3.162,$ find to three places of decimal, the value of the following:
$\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$
→$\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that$\text{AC}\bot\text{BD}.$ Prove that PQRS is a rectangle.
→Draw, in the same diagram, a histogram and a frequency polygon to represent the following data which shows the monthly cost of living index of a city in a period of 2 years:
| Cost of living inex: | 440-460 | 460-480 | 480-500 | 500-520 | 520-540 | 540-560 | 560-580 | 580-600 |
| No.of months: | 2 | 4 | 3 | 5 | 3 | 2 | 1 | 4 |
Construct a right-angled triangle ABC whose base BC is 6cm and the sum of the hypotenuse AC and other side AB is 10cm.
Find the length of a chord which is at a distance of 5cm from the centre of a circle of radius 10cm.
30 circular plates, each of radius 14cm and thickness 3cm are placed one above the another to form a cylindrical solid. Find:
- The total surface area.
- Volume of the cylinder so formed.
Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.
Thirty children were asked about the number of hours they watched TV programes in the previous week. The results were found as follows:
1, 6, 2, 3, 5, 12, 5, 8, 4, 8, 10, 3, 4, 12, 2, 8, 15, 1, 17, 6, 3, 2, 8, 5, 9, 6, 8, 7, 14, 2
- Make a frequency distribution table for this data, taking class width 5 and one of the class intervals as 5-10.
- How many children watched television for 15 or more hours a week.
In the given figure, if $\text{AB }||\text{ CD},\text{EF }||\text{ BC},\angle\text{BAC}=65^\circ$ and $\angle\text{DHF}=35^\circ,$ find $\angle\text{AGH}.$
→ABCD is a cyclic trapezium with AD || BC. If $\angle\text{B}=70^\circ,$determine other three angles of the trapezium.
→