Question

Simplify:[(√5+1÷√5-1)+(√5-1÷√5+1)]

Answer 1

GIVEN :

The expression is $(\frac{\sqrt{5}+1}{\sqrt{5}-1})+(\frac{\sqrt{5}-1}{\sqrt{5}+1})$

TO FIND :

The simplification of the given expression.

SOLUTION :

Given that the expression is $(\frac{\sqrt{5}+1}{\sqrt{5}-1})+(\frac{\sqrt{5}-1}{\sqrt{5}+1})$

Now simplify the given expression, that is solve the given expression as below:

$(\frac{\sqrt{5}+1}{\sqrt{5}-1})+(\frac{\sqrt{5}-1}{\sqrt{5}+1})$

$=\frac{(\sqrt{5}+1)(\sqrt{5}+1)+(\sqrt{5}-1)(\sqrt{5}-1)}{(\sqrt{5}-1)(\sqrt{5}+1)}$

By using the power rule of exponents is given by:

$(a+b)^m.(a+b)^m=(a+b)^{m+m}$

$=\frac{(\sqrt{5}+1)^{1+1}+(\sqrt{5}-1)^{1+1}}{(\sqrt{5}-1)(\sqrt{5}+1)}$

$=\frac{(\sqrt{5}+1)^{2}+(\sqrt{5}-1)^{2}}{(\sqrt{5}-1)(\sqrt{5}+1)}$

Here by using the algebraic identities :

i) $(a+b)^2=a^2+2ab+b^2$

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

iii) $(a-b)(a+b)=a^2-b^2$

$=\frac{(\sqrt{5}^2+2\sqrt{5}(1)+1^2)+(\sqrt{5}^2-2\sqrt{5}(1)+1^2}{(\sqrt{5}^2-1^2)}$

$=\frac{(5+2\sqrt{5}+1)+(5-2\sqrt{5}+1)}{5-1}$

By adding the like terms we get that,

$=\frac{12}{4}$

$=3$

∴  $(\frac{\sqrt{5}+1}{\sqrt{5}-1})+(\frac{\sqrt{5}-1}{\sqrt{5}+1})=3$

∴ The simplified form for the given expression  $(\frac{\sqrt{5}+1}{\sqrt{5}-1})+(\frac{\sqrt{5}-1}{\sqrt{5}+1})$ is 3.

Answer 2

Answer:

answer is 3

Step-by-step explanation:

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