Question

Heya,

Prove that 96 - √7 is an irrational number.

[Class 10]

➡ Spam and meaningless answers will be reported ^_^

#Thank_You

Answer 1

Let 96-root7 be a rational
So let it be equal to p/q where p and q are whole no’s
So, 96-root7=p/q
- root7= p/q-96
Root 7 = -p+96q/q
Which contradicts the fact that root 7 is irrational
So it is a irrational no

Answer 2

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$\huge \mathbb\fcolorbox{black}{lavenderblush}{Answer : ♡}$

Let us assume that 96 - √7 is a rational number,

So, we know that all rational numbers can be expressed in the form p/q where q ≠ 0.

$96 - \sqrt{7} = \frac{p}{q} \\ \\ - \sqrt{7} = \frac{p}{q} - 96 \\ \\ - \sqrt{7} = \frac{p - 96q}{q} \\ \\ \sqrt{7} = \frac{96q - p}{q}$

RHS :

$\frac{96q - p}{q}$

is a rational number clearly, because it can be expressed as p/q, But,

LHS :

$\sqrt{7}$

is not a rational number (as we already know that √7 is an irrational number), which is not possible.

(For easy understanding :

Because, a rational number is always equal to a rational number only).

Hence, this contradicts the assumption that 96-√7 is a rational number, means our assumption is wrong.

Therefore, 96-√7 is an irrational number.

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Hope my Answer is correct and useful !