Question

If a = 4 + √15 ,find the value of a - 1/a​

Answer 1

$\Large {\underline { \sf {Answer :}}}$

2√15

$\Large {\underline { \sf {Clarification :}}}$

Here, we are provided with the value of a, that is :

• $\sf {a = 4 + \sqrt{15} }$

We are asked to calculate the value of :

• $\sf {a- \dfrac{1}{a} }$

Step 1 : Firstly we'll write 1/a in terms of a.

Step 2 : After that, we'll rationalise the denominator of it.

Step 3 : After rationalising the denominator, we'll add a and 1/a as usually we perform the subtraction.

How to rationalise the denominator?

• In order to rationalise the denominator of any fraction, we multiply the rationalising factor of the denominator with both the denominator and the numerator of the fraction.

$\Large {\underline { \sf {Explication \; of \; steps :}}}$

Here,

$\longrightarrow \sf {a = 4 + \sqrt{15} }$

Let it be the equation 1.

Therefore,

$\longrightarrow \sf {\dfrac{1}{a} = \dfrac{1}{4 + \sqrt{15}} }$

Rationalising the denominator of the above fraction :

• Rationalising factor of (4+√15) is (4-√15)

Multiplying (4-√15) with both the numerator and the denominator.

$\longrightarrow \sf {\dfrac{1}{a} = \dfrac{1}{4 + \sqrt{15}} \times \dfrac{4 - \sqrt{15}}{4 - \sqrt{15}}}$

$\longrightarrow \sf {\dfrac{1}{a} = \dfrac{1}{4 + \sqrt{15}} \times \dfrac{4 - \sqrt{15}}{4 - \sqrt{15}}}$

$\longrightarrow \sf {\dfrac{1}{a} =\dfrac{1(4 - \sqrt{15})}{(4 + \sqrt{15})(4 - \sqrt{15}) } }$

$\longrightarrow \sf {\dfrac{1}{a} =\dfrac{4 - \sqrt{15}}{(4+ \sqrt{15})(4 - \sqrt{15}) } }$

Now, we know that,

• $\sf{ (a+b)(a-b) = a^2 - b^2}$

$\longrightarrow \sf {\dfrac{1}{a} =\dfrac{4 - \sqrt{15}}{ (4)^2 - (\sqrt{15})^2 } }$

$\longrightarrow \sf {\dfrac{1}{a} =\dfrac{4 - \sqrt{15}}{ 16 - 15 } }$

$\longrightarrow \sf { \dfrac{1}{a} = 4 - \sqrt{15} \dots \mathfrak {(eq.\; 2)}}$

Now,

$\longrightarrow \sf {a- \dfrac{1}{a} }$

Substituting the values from equation 1 and equation 2.

$\longrightarrow \sf {(4+\sqrt{15})- (4 - \sqrt{15})}$

$\longrightarrow \sf {4+\sqrt{15}- 4 + \sqrt{15}}$

$\longrightarrow \sf {\sqrt{15}+ \sqrt{15}}$

$\longrightarrow\underline{ \boxed{\sf {2\sqrt{15}}}} \; \bigstar$

$\therefore$ The required answer is 2√15.