Question

Prove that 5√8-36 is irrational.
I'll mark you as the brilientest .
And give you 50 points.

Answer 1

Let us assume that 5 root8 - 36 is a rational number.

Hence,

It can be expressed in the form of x/y where x and y are integers and y is not equal to 0 and both x and y are co-primes..

Thus,

$5 \sqrt{8} - 36 = \frac{x}{y} \\ \\ \\5.2 \sqrt{2} - 36 = \frac{x}{y} \\ \\ \\10 \sqrt{2} - 36 =\frac{x}{y} \\ \\ \\10 \sqrt{2} = \frac{x}{y} + 36 \\ \\ \\10 \sqrt{2} = \frac{x + 36y}{y} \\ \\ \\ \sqrt{2} = \frac{x + 36y}{10y}$

We already know that root2 is an irrational number while the RHS of the equation is a rational number.

Both can never be equal.

Hence,

It is a contradiction which has arose because we took 5root8-36 as a rational. Number.

Thus,

5root8 is an irrational number.

Answer 2

Step-by-step explanation:

Let us assume that 5 root8 - 36 is a rational number.

Hence,

It can be expressed in the form of x/y where x and y are integers and y is not equal to 0 and both x and y are co-primes..

Thus,

$\begin{lgathered}5 \sqrt{8} - 36 = \frac{x}{y} \\ \\ \\5.2 \sqrt{2} - 36 = \frac{x}{y} \\ \\ \\10 \sqrt{2} - 36 =\frac{x}{y} \\ \\ \\10 \sqrt{2} = \frac{x}{y} + 36 \\ \\ \\10 \sqrt{2} = \frac{x + 36y}{y} \\ \\ \\ \sqrt{2} = \frac{x + 36y}{10y}\end{lgathered}$

We already know that root2 is an irrational number while the RHS of the equation is a rational number.

Both can never be equal.

Hence,

It is a contradiction which has arose because we took 5root8-36 as a rational. Number.

Thus,

5root8 is an irrational number.