Question

if root5 +root3/2root5-3root3 =a-b root15 then find the value of a and b

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Answer 1

Answer:

a = -19/7

b = 5/7

Step-by-step explanation:

As per the provided information in the given question, we have :

• $\sf \dfrac{ \sqrt{5} + \sqrt{3}}{ 2\sqrt{5} - 3 \sqrt{3}} = a - b \sqrt{15}$

We've been asked to calculate the value of a and b. In order to calculate the value of a and b. We'll have to rationalise the denominator of the given fraction.

Rationalisation is nothing but the process of making the denominator. It is done by multiplying the rationalising factor of the denominator with both the numerator and the denominator.

Here, the denominator is (2√5 ― 3√3) which is in the form of (a ― b). The rationalising factor of (a ― b) is (a + b). Henceforth, the rationalising factor of (2√5 ― 3√3) will be (2√5 + 3√3).

So, multiplying (2√5 + 3√3) with both the numerator and the denominator of the fraction.

$\implies \sf { \dfrac{ \sqrt{5} + \sqrt{3}}{ 2\sqrt{5} - 3 \sqrt{3}} \times \dfrac{2\sqrt{5} + 3 \sqrt{3}}{2\sqrt{5} + 3 \sqrt{3}} }$

Now, rearrange the terms so that we can perform the multiplication efficiently and easily.

$\implies \sf { \dfrac{ (\sqrt{5} + \sqrt{3})(2\sqrt{5} + 3 \sqrt{3})}{ (2\sqrt{5} - 3 \sqrt{3})(2\sqrt{5} + 3 \sqrt{3}) } }$

Now, in the numerator we'll perform multiplication using the property of multiplication over addition and in the denominator we will use the algebraic identity that is (a + b)(a ― b) = a² ― b².

$\implies \sf { \dfrac{ \sqrt{5} (2\sqrt{5} + 3 \sqrt{3}) + \sqrt{3}(2\sqrt{5} + 3 \sqrt{3})}{ (2\sqrt{5})^2 - (3 \sqrt{3})^2 } }$

Performing multiplication in the numerator and writing the squares of the numbers in the denominator.

$\implies \sf { \dfrac{ 10 + 3\sqrt{15} + 2\sqrt{15} + 9 }{ 20 - 27 } }$

Now, performing addition in the numerator and subtraction in the denominator.

$\implies \sf { \dfrac{ 19 + 5\sqrt{15} }{ -7} }$

This can be written as,

$\implies \sf { \dfrac{ -(19 + 5\sqrt{15}) }{ -(-7)} }$

$\implies \sf { \dfrac{ -19 - 5\sqrt{15} }{ 7} }$

Or

$\implies \sf { \dfrac{ -19}{7} - \dfrac{ 5\sqrt{15} }{ 7} }$

According to the question,

$\implies \sf \dfrac{ \sqrt{5} + \sqrt{3}}{ 2\sqrt{5} - 3 \sqrt{3}} = a - b \sqrt{15}$

Or,

$\implies \sf { \dfrac{ -19}{7} - \dfrac{ 5\sqrt{15} }{ 7} = a - b \sqrt{15} }$

Comparing both sides, we get that

• a = -19/7
• b = 5/7

$\rule{200}2$