Question

if p - root q = 4+root 2 / 2 + root 2 then (p,q) =

Answer 1

Answer:

(p , q) = (3 , 2)

Step-by-step explanation:

Given :

$\sf p-\sqrt{q}=\dfrac{4+\sqrt{2}}{2+\sqrt{2}}$

To find :

(p , q) = ?

Solution :

Rationalizing factor = 2 - √2

Multiply and divide the given fraction by (2 - √2)

$\sf \dfrac{4+\sqrt{2}}{2+\sqrt{2}} \times \dfrac{2-\sqrt{2}}{2-\sqrt{2}} \\\\\\ \sf =\dfrac{(4+\sqrt{2})(2-\sqrt{2})}{(2+\sqrt{2})(2-\sqrt{2})} \\\\\\ \sf =\dfrac{4(2-\sqrt{2})+\sqrt{2}(2-\sqrt{2})}{2(2-\sqrt{2})+\sqrt{2}(2-\sqrt{2})} \\\\\\ \sf =\dfrac{8-4\sqrt{2}+2\sqrt{2}-\sqrt{2}^2}{4-2\sqrt{2}+2\sqrt{2}-\sqrt{2}^2} \\\\\\ \sf =\dfrac{8-2\sqrt{2}-2}{4-2} \\\\ \sf =\dfrac{6-2\sqrt{2}}{2} \\\\ \sf =3-\sqrt{2}$

Comparing (3 - √2) with p - √q , we get

• p = 3
• q = 2

(p , q) = (3 , 2)

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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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Answer 2

4+root 2 / 2+root 2

= 4+root 2 / 2+root 2 * 2-root2/2-root2

= 4+root 2*2-root2/2+root2*2-root2

= 8-4root2+2root2-root2^2/4-root2^2 because (x+y)(x-y)=x^2-y^2

= 8-2root2-2/4-2

=8-2root2-2/2

=6-2root2/2

=3-root2

=p-rootq

p=3,q=2