Question

classify the following numbers rational or irrational √45​

Answer 1

Answer:

Root 45 is an irrational number

Step-by-step explanation:

It cannot be expressed as p/q

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Answer 2

Answer:

$\sqrt{45}$ = $\sqrt{5*3*3}$

So, It means, $3\sqrt{5}$

Let us assume that $3\sqrt{5}$ is rational.

So ,                  $3\sqrt{5}$ = p/q    [where p, q are integers and q is not equal to zero]

                      $\sqrt{5}$ = p/3q

Now, let us assume, $\sqrt{5}$  = a/b  [where a,b are co-primes]

                                On squaring both the sides,

                              ($\sqrt{5}$ )^2 = (a/b)^2

                               5 = $a^{2}$/$b^{2}$

                              5$b^{2}$ = $a^{2}$ [ This shows that 5 is a factor of a]

Also,  Taking a = divisor ; b = dividend ; c = quotient

                                5$b^{2}$ = (5c$)^{2}$

                                 $b^{2}$ = 5$c^{2}$ [ This shows that 5 is a factor of b]

Thus, a & b have common factor 5, so they are not co-primes and $\sqrt{5}$ is irrational.

Hence, to satisfy $\sqrt{45}$ is irrational.

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