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State true or false: $(4 + 32)$ is irrational.

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True

False

answer is A.

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Detailed Solution

Given is the expression

(4 + 32)

.
We have to check if

(4 + 32)

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

(4 + 32)

is rational.
Then by the closure property of rational numbers, if we add, subtract, multiply or divide it by a rational, the result should be a rational.

(4 + 32) − 4 = 32

So, we can say that

32

is also rational.
Again, we know that product of 2 rational numbers is also a rational number, so we have,

(32) × 13 = 32 × 13 = 2

So, we can say that

2

is also rational.
But, as discussed, both

2

and

32

are irrational as their radicand i.e., 2 is an imperfect square.
So, this contradicts our assumption, therefore, we conclude that

(4 + 32)

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

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