Question

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





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Answer 1

Answer:

Step-by-step explanation:

Given:

→5+2√3÷7+4√3=a + b√3.

Answer:

→Given (5 + 2√3)/(7 + 4√3) = a + b√3

Rationalizing the denominator on left-hand-side by multiplying the numerator and denominator with (7 - 4√3),

→(5 + 2√3) (7 - 4√3)/(7 + 4√3) (7 - 4√3) = a + b√3 ,

Multiply term by term the two expressions on numerator of L.H.S. and for the denominator apply the identity (m+n) (m-n) = m² - n² . We obtain,

→(35 - 20√3 + 14√3 - 8.√3.√3)/[7² - (4√3)²] = a + b√3 ,

→Or, (35 - 6√3 - 8.3)/(49 - 48) = a + b√3 ,

→Or, (35 - 6√3 - 24)/1 = a + b√3 ,

→Or, 11 - 6√3 = a + b√3 ,

→Now equate the rational and irrational terms from both sides.

→11 = a,

→Or, a = 11 ,

→- 6√3 = b√3 ,

⇒ b = -6 ,

Verification:

→To prove (5 + 2√3)/(7 + 4√3) = a + b√3 ,

→i.e. to prove (5 + 2√3) = (a + b√3) (7 + 4√3) ,

→Substituting for a=11 and b=-6,

→R.H.S.= (a + b√3) (7 + 4√3) ,

→(11 - 6√3) (7 + 4√3) = 11.7 + 11.4√3 - 6√3.7 - 6.4.√3.√3 = 77 + 44√3 - 42√3 - 24.3 .

→77 + 2√3 - 72 = 5 + 2√3 = L.H.S.

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Answer 2

Answer:

a=11 b= -6

Step-by-step explanation:

GIVEN,

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

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

$= \frac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3}) }{ {7}^{2} - ({4 \sqrt{3}) }^{2} }$

$= \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 }{49 - 48}$

$= \frac{11 - 6 \sqrt{3}}{1}$

$= 11 - 6 \sqrt{3}$

$\bold{\boxed{therefore, a=11 , b= -6}}$