Question

$if x and y are the rational number and(2 \sqrt{5} \div (2 \sqrt{5} - \sqrt{3} + (2 \sqrt{5} - \sqrt{3 \div (2 \sqrt{5} } + \sqrt{3) } = x + y \sqrt{15} find \: the \: value \: of \: x \: and \: y$
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Answer 1

Correct Question :-

[(2√5 + √3) / (2√5 - √3)] + [(2√5 - √3) / (2√5 + √3)] = a + √15b

To Find :-

• value of a & b ?

Solution :-

Lets First Rationalize The first Part :-

→ [(2√5 + √3) / (2√5 - √3)]

Multiply & Divide by (2√5 + √3) we get,

→ [(2√5 + √3) / (2√5 - √3)] * [ (2√5 + √3) / (2√5 + √3) ]

Numerator Become (a + b)(a+b) = (a + b)² & Denominator Becomes (a + b)(a - b) = a² - b² .

So,

→ [ (2√5 + √3)² ] / [ (2√5)² - (√3)² ]

Using (a + b)² = a² + b² + 2ab in Numerator,

→ [ (20 + 3 + 4√15) / (20 - 3) ]

→ (23 + 4√15) / 17 ----------- Equation (1)

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Rationalize Second Part Now,

→ [(2√5 - √3) / (2√5 + √3)]

Multiply & Divide by (2√5 - √3)

→ [ (2√5 - √3)² / {(2√5)² - (√3)²} ]

→ [ (20 + 3 - 4√15) / 17 ]

→ (23 - 4√15) / 17 ----------- Equation (2)

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Adding Equation (1) & (2) Now, we get,

→ [(23 + 4√15) / 17] + [(23 - 4√15) / 17] = a + √15b

→ [ (23 + 23 + 4√15 - 4√15) / 17 ] = a + √15b

→ (46/17) = a + √15b

Comparing we get,

→ a = (46/17) .

→ b = 0 .