Question

if a and b are rational numbers and 5+2√3/7+4√3=a-b √3 , find the values of a and b.

Answer 1

Answer:

a=11

b=6

it is the answer

Answer 2

Answer:

Concept:

rational numbers:

p/q of two integers with  q ≠ 0 can be used to represent a number as a rational number in mathematics. The set of rational numbers also contains all of the integers, which can each be expressed as a quotient with the integer as the numerator and 1 as the denominator.

Step-by-step explanation:

Given:

$\\\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=a+b \sqrt{3}$

Find:

The values of a and b

Solution:

         $\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=a+b \sqrt{3}$

Since rationalization of  $\frac{a+\sqrt{b}}{c+\sqrt{d}}=\frac{a+\sqrt{b}}{c+\sqrt{d}} \times \frac{c-\sqrt{d}}{c-\sqrt{d}}$

          Taking L.H.S.

          By rationalization,  

                                     $\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}} \times \frac{7-4 \sqrt{3}}{7-4 \sqrt{3}}$

Since,   $\left\{(a+b)(a-b)=a^{2}-b^{2}\right\}$

                      $=\frac{(5+2 \sqrt{3}) \times(7-4 \sqrt{3})}{\left[(7)^{2}-(4 \sqrt{3})^{2}\right]}$

                       $&=\frac{35+14 \sqrt{3}-20 \sqrt{3}-8 \sqrt{3} \sqrt{3}}{49-16 \sqrt{3} \sqrt{3}}$

                       $&=\frac{35-\sqrt{3}(14-8)-24}{49-48}$

                        $&=\frac{35-\sqrt{3}(6)-24}{1}$

                        $&=11-6 \sqrt{3}$

Hence, by comparing a = 11 and b = -6

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