Question
In the following determine rational numbers a and b:
$\frac{3+\sqrt2}{3-\sqrt2}=\text{a}+\text{b}\sqrt{2}$
$\frac{3+\sqrt2}{3-\sqrt2}=\text{a}+\text{b}\sqrt{2}$
✓
Answer
We know that rationalization factor for $3-\sqrt2$ is $3+\sqrt2.$ We will multiply numerator and denominator of the given expression $\frac{3+\sqrt2}{3-\sqrt2}$ by $3+\sqrt2,$ to get
$\frac{3+\sqrt2}{3-\sqrt2}\times\frac{3+\sqrt2}{3+\sqrt2}=\frac{(3)^2+\big(\sqrt2\big)^2+2\times3\times\sqrt2}{(3)^2-\big(\sqrt2\big)^2}$
$=\frac{9+2+6\sqrt2}{9-2}$
$=\frac{11+6\sqrt2}{7}$
$=\frac{11}{7}+\frac{6}{7}\sqrt2$
On equating rational and irrational terms, we get
$\text{a}+\text{b}\sqrt2=\frac{11}{6}+\frac{6}{7}\sqrt2$
Hence, we get $\text{a}=\frac{11}{7},\ \text{b}=\frac{6}{7}.$
$\frac{3+\sqrt2}{3-\sqrt2}\times\frac{3+\sqrt2}{3+\sqrt2}=\frac{(3)^2+\big(\sqrt2\big)^2+2\times3\times\sqrt2}{(3)^2-\big(\sqrt2\big)^2}$
$=\frac{9+2+6\sqrt2}{9-2}$
$=\frac{11+6\sqrt2}{7}$
$=\frac{11}{7}+\frac{6}{7}\sqrt2$
On equating rational and irrational terms, we get
$\text{a}+\text{b}\sqrt2=\frac{11}{6}+\frac{6}{7}\sqrt2$
Hence, we get $\text{a}=\frac{11}{7},\ \text{b}=\frac{6}{7}.$
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
If one angle of a triangle is equal to the sum of the other two, show that the triangle is right angled.
In the given figure, BE and CF are two equal altitudes of $\triangle\text{ABC}.$
Show that:
→Show that:
- $\triangle\text{ABE}\cong\triangle\text{ACF},$
- AB = AC.
In figure, AB || CD and P is any point shown in the figure. Prove that: $\angle\text{ABP}+\angle\text{BPD}+\angle\text{CDP}=360^\circ$
→AD is a median of the $\triangle\text{ABC}.$ Is it true that AB + BC + CA > 2AD? Give reason for your answer.
→Evaluate:
$\Big[5\Big(8^{\frac{1}{3}}+27^{\frac{1}{3}}\Big)^3\Big]^{\frac{1}{4}}$
→$\Big[5\Big(8^{\frac{1}{3}}+27^{\frac{1}{3}}\Big)^3\Big]^{\frac{1}{4}}$
If $\text{x}^2+\frac{1}{\text{x}^2}=66,$ find the value of $\text{x}-\frac{1}{\text{x}}.$
→Give the geometric representations of the following equations:
- On the number line.
- On the Cartesain plane.
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord
Find five rational numbers between $\frac{3}{5}$ and $\frac{2}{3}.$
→In figure, state Which lines are parallel and why.
