Question

if 2+√3 by2-√3=a+b√3 than find the value of a&b.​ please give me answer

Answer 1

Answer:

a = 7 , b = 4

Step-by-step explanation:

$\sf \dfrac{2+\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$

Rationalizing factor = 2 + √3

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

 $\sf =\dfrac{2+\sqrt{3}}{2-\sqrt{3}} \\\\\\ \sf =\dfrac{2+\sqrt{3}}{2-\sqrt{3}} \times \dfrac{2+\sqrt{3}}{2+\sqrt{3}} \\\\\\ \sf =\dfrac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})} \\\\\\ \sf =\dfrac{2(2+\sqrt{3})+\sqrt{3}(2+\sqrt{3})}{2(2+\sqrt{3})-\sqrt{3}(2+\sqrt{3})} \\\\\\ \sf =\dfrac{4+2\sqrt{3}+2\sqrt{3}+\sqrt{3}^2}{4+2\sqrt{3}-2\sqrt{3}-\sqrt{3}^2} \\\\\\ \sf =\dfrac{4+4\sqrt{3}+3}{4-3} \\\\\\ \sf =\dfrac{7+4\sqrt{3}}{1} \\\\\\ \sf =7+4\sqrt{3}$

Comparing (7 + 4√3) with (a + b√3)

a = 7 and b = 4

_____________________

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

_______________________

Answer 2

${\large{\textsf{\textbf{\red{Answer}}}}}$

a = 7 , b = 4

${\large{\textsf{\textbf{\green{Step-by-step \: explanation}}}}}$

$\sf \dfrac{2+\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$

• Rationalizing factor = 2 + √3
• Multiply and divide the given fraction by (2 + √3)

 

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

$\sf =\dfrac{2+\sqrt{3}}{2-\sqrt{3}} \times \dfrac{2+\sqrt{3}}{2+\sqrt{3}}$

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

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

$\sf =\dfrac{4+2\sqrt{3}+2\sqrt{3}+\sqrt{3}^2}{4+2\sqrt{3}-2\sqrt{3}-\sqrt{3}^2}$

$\sf =\dfrac{4+4\sqrt{3}+3}{4-3}$

$\sf =\dfrac{7+4\sqrt{3}}{1}$

$\sf =7+4\sqrt{3}$

Comparing (7 + 4√3) with (a + b√3)

a = 7 and b = 4

_____________________