Model problem-2: Proof of 5√2+7√3 is an irrational number-10th class mathematics- Proof of Irrational number

Math for class 10

Before going to learn the proof of 5√2+7√3 is irrational, we know the following:

Every rational number can be written in the

form of p/q (Where p, q are integers and q ≠ 0)

        

        (a + b)² = a² + 2ab + b²

        

        (a - b)² = a² - 2ab + b²

Transposing a term/factor from one side to other

Transpose “+a” to LHS/RHS, it changed to “-a”

Transpose “-a” to LHS/RHS, it changed to “+a”

Transpose “ x a” to LHS/RHS, it changed to “÷ a”

Transpose “÷ a” to LHS/RHS, it changed to “ x a”

Transpose “x  a/b” to LHS/RHS, it changed to “x  b/a”

Simplifications of fractions

For Example:

  

 2/3 + 5 = (2+5 x 3)/3 = (2 +15)/3 = 17/3

 

 7 – 5/4 = (7 x 4 – 5)/4 = (28 – 5)/4 = 23/4

 3/5 + 4/7 = (3x7+4x5)/5x7 =(21+ 20)/35=41/35

 And see the post of "proof of √2 is an irrational" 

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    Prove that 5√2 +7√3 is an irrational.

(We can prove this by contradiction method)

Proof:

Assume that 5√2 + 7√3 is not an irrational number

So 5√2 + 7√3  is a rational number

∴ 5√2 + 7√3 = p/q (p, q are integers and q ≠ 0)

Transpose "+7√3" is to RHS

5√2 = P/q - 7√3

Squaring on both sides, we get

(5√2)² = (P/q – 7√3 )²   

 [Apply (a - b)² = a² - 2ab + b²]

5²(√2)²= (P/q)² – 2 x (p/q) x ((7√3) + (7√3)²

      [Here a= p/q and b=7√3 ]

25(2) = (P²/q²) – (14√3)(p/q) + 7²(√3)²

50 = (P²/q²) – (14p/q)(√3)+ 49(3)

Transpose “– (14p/q)(√3)” to LHS

50 + (14p/q)(√3)= P²/q² + 147

Transpose “+50” to RHS

(14p/q)(√3)= P²/q² + 147 - 50

(14p/q)(√3)= P²/q² + 97

    Multiply 97 by q²

(14p/q)(√3) = (p² + 97q²)/q²  

Transpose “ x (14p/q)” to RHS

(√3) = (p² + 97q²)/q²  x (q/14p)

(√3) = (p² + 97q²)/14pq (“q” cancelled)

Since p and q are integers,

“(p² + 97q²)/14pq” is a rational number

But √3 is an irrational number

 This is a contradiction

This contradiction is arisen due to our assumption

So our assumption is wrong

“ 5√2 + 7√3 is not an irrational” is wrong

Therefore 5√2 + 7√3 is an irrational

        Hence proved

Solve the following model problems for Practice

1. Prove that 2√3 + 3√7 is an irrational number

2. Prove that 2√7 - √5 is an irrational number

3. Prove that √5- √2 is an irrational number