Question

i wanted to know how it came, help the kid~

$\frac{1}{2} \times \frac{ \sqrt{3} }{2} + 2 = \frac{ \sqrt{2} }{ \sqrt{2} \times \sqrt{2} } \times \frac{ \sqrt{3} \times ( \sqrt{3} - 1) }{2( \sqrt{3} + 1)( \sqrt{3} - 1)}$

i just need the explanation about it, no need to solve it further, and if u r so helping person than u can solve it.

Mainly i am asking this to get how we got that thang after that equal through the small expression.​

Answer 1

Answer:

Here,

LHS≠RHS (ON SOLVING)

So, there maybe slight mistakes in the expression given.

yes...You can even verify using the calculator.

On solving LHS:

$= \frac{1}{2} \times \frac{ \sqrt{3} }{2} + 2$

Using DMAS,

$= \frac{ \sqrt{3} }{4} + 2$

$LHS = \frac{ \sqrt{3} + 8}{4}$

On solving RHS:

$= \frac{ \sqrt{2} }{ \sqrt{2} \times \sqrt{2} } \times \frac{ \sqrt{3}( \sqrt{3 } - 1) }{2( \sqrt{3 } - 1)( \sqrt{3 } + 1) }$

$= \frac{ \sqrt{2} }{2} \times \frac{ \sqrt{3} }{2} \times \frac{( \sqrt{3} - 1) }{( \sqrt{3} - 1)( \sqrt{3} + 1) }$

$= \frac{ \sqrt{3} }{4} \times \sqrt{2} \times \frac{1}{ \sqrt{3} + 1 }$

$= \frac{ \sqrt{3} }{4} \times \frac{ \sqrt{2} }{ \sqrt{3} + 1 }$

On rationalising the denominator:

$\frac{ \sqrt{3} }{4} \times \frac{ \sqrt{6} - \sqrt{2} }{2}$

$= \frac{ \sqrt{18} - \sqrt{6} }{8}$

$RHS = \frac{3 \sqrt{2} - \sqrt{6} }{8}$

So, LHS≠RHS

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BUT STILL, LET ME EXPLAIN WHAT HAS ACTUALLY HAPPENED HERE,

which gave you a way big expression from a small one.

As, we know that, Multiplying same thing on both RHS and LHS gives the same & doesn't affect the actual solution. (as per an Axiom read in class 9)

E.g.

$If .\: x = 2$

Then on multiplying by 2 on both sides:

$2x = 2 \times 2$

or

$x = \frac{4}{2} = 2$

(On shifting '2' to RHS)

Conclusion: On multiplying and dividing an expression by the same thing on both Numerator and Denominator doesn't affect the actual solution.

So, you can multiply and divide an expression with any expression as it won't affect the actual solution.

E.g. "(√3-1)" is multiplied and divided with the numerator and denominator in LHS.

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So, if there are few missing expressions in the RHS,

THEN THE CORRECT EXPRESSION WHICH COULD SATISFY THE LHS can be:

$\frac{ \sqrt{2} }{ \sqrt{2} \times \sqrt{2} } \times \frac{ \sqrt{3} ( \sqrt{3} - 1)}{2( \sqrt{3} - 1)( \sqrt{3} + 1) } \times ( \frac{ \sqrt{3} + 1}{ \sqrt{2} } \times \frac{ \sqrt{3} + 8 }{ \sqrt{3} } )$

Although there can be many such expressions, where same thing is multiplied and divided by both numerator and denominator.