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
A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
In figure, ray $OS$ stand on a line $POQ$. Ray $OR$ and ray $OT$ are angle bisectors of $\angle\text{POS}$ and $\angle\text{SOQ}$ respectively. If $\angle\text{POS}=\text{x},$ find $\angle\text{ROT}.$
→$BD$ is one of the diagonals of a quadrilaterl $ABCD$.
If $\text{AL}\perp\text{BD}$ and $\text{CM}\perp\text{BD},$show that
ar$($quadrilaterl $ABCD)$ $=\frac{1}{2}\times\text{BD}\times(\text{AL}+\text{CM}).$
→If $\text{AL}\perp\text{BD}$ and $\text{CM}\perp\text{BD},$show that
ar$($quadrilaterl $ABCD)$ $=\frac{1}{2}\times\text{BD}\times(\text{AL}+\text{CM}).$
Find a rational number and also an irrational number lying between the numbers $0.3030030003...$ and $0.3010010001...$
→In the given figure, $BA || ED$ and $BC || EF$. Show that $\angle\text{ABC}=\angle\text{DEF}.$
→ABCD is a cyclic trapezium with AD || BC. If $\angle\text{B}=70^\circ,$determine other three angles of the trapezium.
→E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Prove that EF || AB and$\text{EF}=\frac{1}{2}(\text{AB}=\text{CD})$
[Hint: Join BE and produce it to meet CD produced at G.]
→[Hint: Join BE and produce it to meet CD produced at G.]
ABCD is a rhombus, EABF is a straight line such that EA = AB = BF. Prove that ED and FC when produced meet at right angles.
A cylindrical water tank of diameter $1.4\ m$ and height $2.1\ m$ is being fed by a pipe of diameter $3.5\ cm$ through which water flows at the rate of $2$ metre per second. In how much time the tank will be filled?
→In figure, Line $l_1$ and $l_2$ intersect at $O$, forming angles as shown in the figure. If $x = 45$, find the value of $y, z$ and $u.$ 
→