Question

prove that 5 root 3 minus 2 root 7 is an irrational number

Answer 1

Step-by-step explanation:

assuming  5 root 3 - 2 root 7 to be a rational number,

we can say,

5 root 3 - 2 root 7 = p/q (where p and q are co-prime numbers)

squaring both sides, we get,

=> (5 root 3 - 2 root 7)^2 = (p/q)^2    (using the identity {a - b}^2)

=>75 - 20 root 21 + 28 = p^2/q^2

=>103 - 20 root 21 = p^2/q^2

=> root 21 = (p^2 -103q^2)/-20q^2 ----- (1)

according to (1), root 21 is a rational number. However, we know that it is an irrational number. Since, our above outcome was wrong, we can say that our first assumption i.e, 5 root 3 - 2 root 7 to be a rational number is false. Hence, 5 root 3 - 2 root 7 is an irrational number.

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