Question

prove that 4-5root 2 is irrational

Answer 1

Answer:

here is ur answer

Step-by-step explanation:

Let us assume that (4−52) is rational .

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.4−(4−52)is rational

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.4−(4−52)is rational⇒52 is rational

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.4−(4−52)is rational⇒52 is rationalwhich is only possible if 5 is rational and 2 is rational.

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.4−(4−52)is rational⇒52 is rationalwhich is only possible if 5 is rational and 2 is rational.As we know prouduct of two rational number s rational

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.4−(4−52)is rational⇒52 is rationalwhich is only possible if 5 is rational and 2 is rational.As we know prouduct of two rational number s rationalBut the fact is 2 is an irrational.

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.4−(4−52)is rational⇒52 is rationalwhich is only possible if 5 is rational and 2 is rational.As we know prouduct of two rational number s rationalBut the fact is 2 is an irrational.Which is contradictory to our assumption.

Let us assume that (4−52) is rational .Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.4−(4−52)is rational⇒52 is rationalwhich is only possible if 5 is rational and 2 is rational.As we know prouduct of two rational number s rationalBut the fact is 2 is an irrational.Which is contradictory to our assumption.Hence, 4−52 is irrational . hence proved.

Answer 2

To prove : 4 - 5√2 is irrational

:-:-:-: SOLUTION :-:-:-:

Assume that 4 - 5√2 is a rational number.

Then, It can be expressed as a/b where a & b is a pair of co-prime numbers.

So,

$\rm \frac{a}{b} = 4 - 5 \sqrt{2} \\ \\ = > \frac{a}{b} - 4 = \: \: - 5 \sqrt{2} \\ \\ = > \frac{ \frac{a}{b} - 4}{ - 5} = \sqrt{2} \\ \\ = > \frac{ a - 4b}{ - 5b} = \sqrt{2} \\ \\ = > \frac{ - a + 4b}{5b} = \sqrt{2} \\ \\ = > \frac{4b - a}{5b} = \sqrt{2}$

Here, LHS will result as an integer but RHS is an irrational number.

This implies that,

$\bf{}LHS \neq RHS$

And, the obtained result defies the law of an algebraic expression which states that LHS = RHS in each case of any equation.

Thus, Our assumption was wrong that 4 -5√2 is a rational number.

Hence, It is concludes that 4 -5√2 is a irrational number.