Question

if b=(2-√3) , find b-1/b​

Answer 1

Answer:

b - 1/b = −2√3

Step-by-step explanation:

Given :

b = (2 - √3)

To find :

the value of b - 1/b

Solution :

First, we have to find the value of 1/b then by subtracting the value of 1/b from b , we get the required answer.

Finding the value of 1/b,

1/b = 1/(2 - √3)

The denominator is irrational. Hence we have to rationalize the denominator to get the value of 1/b.

Rationalizing factor = (2 + √3)

Multiply and divide the fraction by (2 + √3)

     $\tt \dfrac{1}{b}=\dfrac{1}{2-\sqrt{3}} \\\\ \tt \dfrac{1}{b}=\dfrac{1}{2-\sqrt{3}} \times \dfrac{2+\sqrt{3}}{2+\sqrt{3}} \\\\ \tt \dfrac{1}{b}=\dfrac{2+\sqrt{3}}{(2-\sqrt{3})(2+\sqrt{3})} \\\\ \tt \dfrac{1}{b}=\dfrac{2+\sqrt{3}}{2(2+\sqrt{3})-\sqrt{3}(2+\sqrt{3})} \\\\ \tt \dfrac{1}{b}=\dfrac{2+\sqrt{3}}{4+2\sqrt{3}-2\sqrt{3}-\sqrt{3}^2} \\\\ \tt \dfrac{1}{b}=\dfrac{2+\sqrt{3}}{4-3} \\\\ \tt \dfrac{1}{b}=\dfrac{2+\sqrt{3}}{1} \\\\ \tt \longrightarrow \dfrac{1}{b}=2+\sqrt{3}$

b - 1/b :

⇒ (2 - √3) - (2 + √3)

⇒ 2 - √3 - 2 - √3

⇒ -2√3

∴ The required value is −2√3

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#Know more :

★ Rationalizing factor :

⇒ The factor of multiplication by which rationalization is done, is called as rationalizing factor.

⇒ If the product of two surds is a rational number, then each surd is a rationalizing factor to other.

⇒ To find the rationalizing factor,  

    =>  If the denominator contains 2 terms, just change the sign between the two terms.

        For example, rationalizing factor of (3 + √2) is (3 - √2)

   => If the denominator contains 1 term, the radical found in the denominator is the factor.

        For example, rationalizing factor of √2 is √2

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