Question
If $\text{x}=3+2\sqrt{2},$ check whether $\text{x}+\frac{1}{\text{x}}$ is rational or irrational.
✓
Answer
$\text{x}=3+2\sqrt{2}$$\Rightarrow\text{x}+\frac{1}{\text{x}}=3+2\sqrt{2}+\frac{1}{3+2\sqrt{2}}$
$=3+2\sqrt{2}+\frac{1}{3+2\sqrt{2}}\times\frac{3-2\sqrt{2}}{3-2\sqrt{2}}$
$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{3^2-\big(2\sqrt{2}\big)^2}$
$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{9-8}$
$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{1}$
$=3+2\sqrt{2}+3-2\sqrt{2}$
$=6$
Thus, the given number is rational.
$=3+2\sqrt{2}+\frac{1}{3+2\sqrt{2}}\times\frac{3-2\sqrt{2}}{3-2\sqrt{2}}$
$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{3^2-\big(2\sqrt{2}\big)^2}$
$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{9-8}$
$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{1}$
$=3+2\sqrt{2}+3-2\sqrt{2}$
$=6$
Thus, the given number is rational.
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 the given figure, two parallel line l and m are intersected by two parallel lines p and q.
Show that $\triangle\text{ABC}\cong\triangle\text{CDA}.$
→Show that $\triangle\text{ABC}\cong\triangle\text{CDA}.$
Following $10$ observations are arranged in ascending order as follows. $2, 3 , 5 , 9, x + 1, x + 3, 14, 16, 19, 20.$ If the median of the data is $11,$ find the value of $x.$
→Factorize:
$4(x - y)^2 - 12(x - y)(x + y) + 9(x + y)^2$
→$4(x - y)^2 - 12(x - y)(x + y) + 9(x + y)^2$
In figure, ABCD and PQRC are rectangles and Q is the mid-point of AC. Prove that
→- DP = PC
- $\text{PR}=\Big(\frac{1}{2}\Big)\text{AC}$
Find the value.
i. | 15 – 2|
ii. | 4 – 9|
iii. | 7| x | -4|
i. | 15 – 2|
ii. | 4 – 9|
iii. | 7| x | -4|
Radius of a circle with centre $O$ is $41$ units. Length of a chord PQ is $80$ units, find the distance of the chord from the centre of the circle.
Given: In a circle with centre O ,
$O P$ is radius and PQ is its chord,
seg $OM \perp$ chord PQ , $P - M - Q$
$O P=41$ units, $P Q=80$ units,
To Find: Distance of the chord from the centre of the circle(OM)
→Given: In a circle with centre O ,
$O P$ is radius and PQ is its chord,
seg $OM \perp$ chord PQ , $P - M - Q$
$O P=41$ units, $P Q=80$ units,
To Find: Distance of the chord from the centre of the circle(OM)
If $3^\text{x}=5^\text{y}=(75^\text{z}),$ show that $\text{z}=\frac{\text{xy}}{2\text{x}+\text{y}}.$
→A cylindrical tub of radius 16cm contains water to a depth of 30cm. A spherical iron ball is dropped into the tub and thus level of water is raised by 9cm. What is the radius of the ball?
The external dimensions of a closed wooden box are $48cm, 36cm, 30cm$. The box is made of 1.5cm thick wood. How many bricks of size $6cm \times 3cm \times 0.75cm$ can be put in this box?
→Simplify:$\frac{2\sqrt{30}}{\sqrt{6}}-\frac{3\sqrt{140}}{\sqrt{28}}+\frac{\sqrt{55}}{\sqrt{99}}$
→