Question

16. Prove that 4-5√2 is an irrational number .​

Answer 1

$\large{\pmb{\sf{\underline{Required\: Solution...}}}}$

The question says that we need to prove that 4 - 5√2 is a Irrational number. The question is mentioned below:

$\pmb{\sf{ 4 - 5 \sqrt{2} }}$

Explanation:

We nee to reduce the number as much as possible since the resultant must be proved that it is irrational.

How to prove that the question is Irrational? By reducing the it! We will assume 4 - 5√2 as x, Then we will use laws of exponents wherever needed and solve it until we can differentiate whether it is irrational or rational!

Step by step explanation:

Assuming 4 - 5√2 as x,

$\implies \sf x = 4 - 5 \sqrt{2}$

Squaring on both sides,

$\implies \sf {x}^{2} =( {4 - 5 \sqrt{2}) }^{2}$

The right hand side seem to be like (a - b)², We will use the law of exponent i.e (a - b)² = a² + b² - 2ab,

${\implies \sf {x}^{2} = {4}^{2} + {5 \sqrt{2} }^{2} - 2(8)(5 \sqrt{2)}}$

Doing the required calculation,

${\implies \sf {x}^{2} = 16 + 50 - 2(8)(5\sqrt{2)}}$

${\implies \sf {x}^{2} =16 + 50 -80( \sqrt{2)}}$

${\implies \sf {x}^{2} = 66 - 80( \sqrt{2)}}$

${\implies \sf \dfrac{{x}^{2} - 66}{ - 80} =( \sqrt{2)}}$

Here, The value in the left hand side is rational, And the value in right hand side is irrational. This directly means that 4 - 5√2 is a Irrational number.

Henceforth, 4 - 5√2 is a Irrational number. Hence, Proved!

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Additional information:

Rational numbers: The numbers which can be written in p/q form. where q ≠ 0 i.e., q is not equal to zero. For example - 7/8, 9/8, 567/738.

Irrational numbers: Irrational numbers are the totally opposite to rational numbers. They cannot be expressed in the form of p/q. The best examples for irrational numbers is $\pmb{{\sf{\pi}}}$ and $\pmb{{\frak{e}}}$