Question

Q. It is being given that
√3 = 1.732 , √5 = 2.236
√6 = 2.449 and √10 = 3.162
•find to three places of decimal , the value of 3+√5 upon 3-√5

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Answer 1

$\huge{\text{Question}}$

Find the value of $\dfrac{3+\sqrt{5} }{3-\sqrt{5} }$ to three places of decimal.

$\huge{\text{Given}}$

$\sqrt{3} =1.732,\sqrt{5} =2.236,\sqrt{6} =2.449,\sqrt{10} =3.162$

$\huge{\text{Solution}}$

Let's calculate $\dfrac{3+\sqrt{5} }{3-\sqrt{5} }$. What do we get if we substitute the value? We have $\dfrac{3+2.236}{3-2.236}$, which is hard to calculate.

However, there a method called rationalization that makes calculation easier.

Let's try it. If we apply an algebraic identity to the denominator, it will be rationalized.

$\text{Required Identity}$

• $a^2-b^2=(a+b)(a-b)$
• $(a+b)^2=a^2+2ab+b^2$

$\text{Given value:}\ \dfrac{3+\sqrt{5} }{3-\sqrt{5} }$

$=\dfrac{3+\sqrt{5} }{3-\sqrt{5} }\times \dfrac{3+\sqrt{5} }{3+\sqrt{5} }$

$=\dfrac{(3+\sqrt{5} )(3+\sqrt{5} )}{(3+\sqrt{5} )(3-\sqrt{5} )}$

Applying the first identity,

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

Applying the second identity,

$=\dfrac{3^2+2\times 3\times \sqrt{5} +(\sqrt{5} )^2}{9-5}$

$=\dfrac{9-6\sqrt{5} +5}{4}$

$=\dfrac{14-6\sqrt{5} }{4}$

$=\dfrac{7-3\sqrt{5} }{2}$

The denominator is a rational number now. So, we can calculate it easily.

$=\dfrac{7-3\times 2.236}{2}$

$=\dfrac{7-6.708}{2}$

$=\dfrac{0.292}{2}=\boxed{0.146}$

$\huge{\text{Required Answer}}$

The answer is $0.146$.

Answer 2

Given :-

√3 = 1.732 , √5 = 2.236

√6 = 2.449 and √10 = 3.162

To Find :-

find to three places of decimal , the value of 3+√5 upon 3-√5

Solution :-

So,

the equation becomes

$\sf \dfrac{3+\sqrt{5}}{3-\sqrt{5}}$

On putting the given vakue of sqrt 5

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

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

$\sf \dfrac{(3+\sqrt{5})^2}{3^2-\sqrt{5}^2}$$\sf 6.854$

$\sf\dfrac{9 + 6\sqrt{5}+5}{9-5}$

$\sf\dfrac{14+6\sqrt{5}}4$

$\sf \dfrac{14 + 6 (2.236)}{4}$

$\sf \dfrac{14 +13.416}{4}$

$\sf\dfrac{27.416}{4}$

$\sf 6.854$