Question
Find rational numbers $a$ and $b$ such that: $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=\text{a}+\text{b}\sqrt{3}$
✓
Answer
$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=\text{a}+\text{b}\sqrt{3}$
we have, $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$ $=\frac{5+2\sqrt{3}}{7+4\sqrt{3}}\times\frac{7-4\sqrt{3}}{7-4\sqrt{3}}$
$=\frac{5\times7-5\times4\sqrt{3}+2\sqrt{3}\times7-2\sqrt{3}\times4\sqrt{3}}{(7)^2-\big(4\sqrt{3}\big)^2}$
$=\frac{35-20\sqrt{3}+14\sqrt{3}-8\times3}{49-16\times3}$
$=\frac{35-6\sqrt{3}-24}{49-48}$ $=\frac{11-6\sqrt{3}}{1}$
$=11-6\sqrt{3}$
Now, $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=\text{a}+\text{b}\sqrt{3}$
$\Rightarrow\text{a}=11$ and $\text{b}=-6$
we have, $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$ $=\frac{5+2\sqrt{3}}{7+4\sqrt{3}}\times\frac{7-4\sqrt{3}}{7-4\sqrt{3}}$
$=\frac{5\times7-5\times4\sqrt{3}+2\sqrt{3}\times7-2\sqrt{3}\times4\sqrt{3}}{(7)^2-\big(4\sqrt{3}\big)^2}$
$=\frac{35-20\sqrt{3}+14\sqrt{3}-8\times3}{49-16\times3}$
$=\frac{35-6\sqrt{3}-24}{49-48}$ $=\frac{11-6\sqrt{3}}{1}$
$=11-6\sqrt{3}$
Now, $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=\text{a}+\text{b}\sqrt{3}$
$\Rightarrow\text{a}=11$ and $\text{b}=-6$
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
In figure, $X$ and $Y$ are the mid points of $AC$ and $AB$ respectively, $QP || BC$ and $CYQ$ and $BXP$ are straight lines. Prove that $\text{ar}(\triangle\text{ABP})=\text{ar}(\triangle\text{ACQ}).$
→In the given figure, if $\text{AB }||\text{ DE}$ and $\text{BD }||\text{ FG}$ such that $\angle\text{FGH}=125^\circ$ and $\angle\text{B}=55^\circ,$ find $x$ and $y.$
→In a triangle $ABC$, if $AB = AC$ and $AB$ is produced to $D$ such that $BD = BC$, find $\angle\text{ACD}:\angle\text{ADC}.$
→Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.
How many metres of cloth, $2.5m$ wide, will be required to make a conical tent whose base radius is $7m$ and height $24m?$ $\Big(\text{Use}\ \pi=\frac{22}{7}\Big).$
→Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 laborers working in a factory, taking one of the class intervals as $210-230 (230$ not included$)$.$ 220, 268, 258, 242, 210, 268, 272, 242, 311, 290, 300, 320, 319, 304, 302, 218, 306, 292, 254, 278, 210, 240, 280, 316, 306, 215, 256, 236$.
→If $2x + 3y = 8$ and $xy = 2$, find the value of $4x^2 + 9y^2.$
→Find the lateral surface area of a petrol storage tank is $4.2m$ in diameter and $4.5m$ high. How much steel was actually used, if $\frac{1}{12}$ of the steel actually used was wasted in making the closed tank?
→Given, $\sqrt{2}=1.414$ and $\sqrt{6}=2.449,$ find the value of $\frac{1}{\sqrt{3}-\sqrt{2}-1}$ correct to $3$ places of decimal.
→Calculate the value of $x$ in the following figures.
→