Question

prove tha root 5 is a irrational


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Answer 1

$\textit{\large{\underline{\underline{\pink{Step by step Explanation:-}}}}}$

Suppose that,,$\mathbf{\large{ \sqrt{5}=\frac{a}{b} }}$__(1)is a rational number ,In which both a and b are integers and Co-prime and $\mathbf{{\large b\ne {0}} }$..

$\mathrm{\large{a=b \sqrt{5}}}$ {On arranging eq.(1)}

$\mathrm{\large{\implies{a^{2}=(b \sqrt{5})^{2}}}}$ {on squaring both sides}

$\mathrm{\large{\implies{a^{2}=5b^{2}}}}$

since,,$\mathrm{\large{{a^{2}}}}$ is divided by 5 ,,so,

a will also be divided by 5.

again,,suppose that,,

$\mathrm{\large{a=5c}}$

$\mathrm{\large{\implies{a^{2}=(5c)^{2}}}}${on squaring both sides}

$\mathrm{\large{\implies{5b^{2}=25c^{2}}}}$ since,$\mathrm{\large{a^{2}=5b^{2}}}$

$\mathrm{\large{\implies{b^{2}=\frac{25c^{2}}{5}}}}$

$\mathrm{\large{\implies{b^{2}=5c^{2}}}}$

since,,$\mathrm{\large{{b^{2}}}}$ is divided by 5 ,,so,

b will also be divided by 5.since a and b both are devided by 5. This seems to contradict the fact that both a and b are co-prime.

Hence root 5 is an irrational number.

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