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State true or false: $(23 − 1)$ is an irrational number.

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True

False

answer is A.

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Detailed Solution

Given is the expression

(23 − 1)

.
We have to check if

(23 − 1)

is irrational or not.
To do so, we will use the method of contradiction and assume that

(23 − 1)

is rational.
Now, let

x = 23 − 1

.
So, we will rewrite x as a fraction as follows,

x + 1 = 23 x + 1 2 = 3

Now as we have assumed x is a rational number, so

x + 1 2

will also be a rational number.
This implies that

3

is also a rational number, which is a contradiction as it is an irrational number.
This means that the assumption we made at the start i.e.,

(23 − 1)

is a rational number, is incorrect.
Thus,

(23 − 1)

is an irrational number.
Hence, the statement is true.
So, the correct option is 1.

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