Question

If a=2+√3 then find a²+1/a²

Answer 1

Given,

The value of a = 2+√3

To Find,

The value of a²+1/a²

Solution,

To calculate the value of a²+1/a², first, we have to find the value of 1/a.

1/a = 1/(2+√3)

Now, rationalizing the denominator,

1/a = 1/(2+√3) *(2-√3)/(2-√3)

1/a = (2-√3)/(2²-√3²)

1/a = (2-√3)

Now,

a²+1/a² = (2+√3)²+(2-√3)²

            = 4+3+4√3+4+3-4√3

            = 14

Hence, the value of a²+1/a² is 14.

Answer 2

$a^{2}+\dfrac{1}{a^{2}}=14$

Step-by-step explanation:

Given, $a=2+\sqrt{3}$

So, $\dfrac{1}{a}=\dfrac{1}{2+\sqrt{3}}$

$=\dfrac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}$

• Hint: We rationalize the denominator by multiplying both the numerator and the denominator by $(2-\sqrt{3})$.

$=\dfrac{2-\sqrt{3}}{(2)^{2}-(\sqrt{3})^{2}}$

$=\dfrac{2-\sqrt{3}}{4-3}$

$=2-\sqrt{3}$

Now, $a+\dfrac{1}{a}$

$=2+\sqrt{3}+2-\sqrt{3}$

$=4$

So, $a^{2}+\dfrac{1}{a^{2}}$

$=(a+\dfrac{1}{a})^{2}-2\times a\times \dfrac{1}{a}$

$=(4)^{2}-2$

$=16-2$

$=14$

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