$(- 2 - \sqrt{3}) (- 2 + \sqrt{3})$ when simplified is
(a) positive and irrational
(b) positive and rational
(c) negative and irrational
(d) negative and rational

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Solution

$(- 2 - \sqrt{3}) (- 2 + \sqrt{3}) = - (2 + \sqrt{3}) \times [- (2 - \sqrt{3})] = (2 + \sqrt{3}) (2 - \sqrt{3})$
$= 2^{2} - {(\sqrt{3})}^{2} [(a + b) (a - b) = a^{2} - b^{2}] = 4 - 3 = 1$
Thus, the given expression when simplified is positive and rational.

Hence, the correct answer is option (b).

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Similar questions

(6 + \sqrt{27}) - (3 + \sqrt{3}) + (1 + 2 \sqrt{3})

when simplified is
(a) positive and irrational
(b) positive and rational
(c) negative and irrational
(d) negative and rational

$(- 2 - \sqrt{3}) (- 2 + \sqrt{3})$ when simplified is

(a) positive and irrational (b) positive and rational (c) negative and irrational (d) negative and rational

Question 8
The product of a non – zero rational and an irrational number is

A) Always irrational
B) Always rational
C) Rational or irrational
D) One

Q. Which of the following options is correct?
(1) Every integer and fraction is a rational number.
(2) A rational number

\frac{p}{q}

is positive if p and q are either both positive or both negative.

(3) A rational number

\frac{p}{q}

is negative if one of p and q is positive and other is negative.
(4) If there are two rational numbers with same denominator, then the one with the larger numerator is larger than the other.

Q. Which of the following options hold?
(1) Every integer is a rational number and every fraction is a rational number.

(2) A rational number

\frac{p}{q}

is positive if p and q are either both positive or both negative.

(3) A rational number

\frac{p}{q}

is negative if one of p and q is positive and other is negative.

(4) If there are two rational numbers with common denominator then the one with the larger numerator is larger than the other.

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-2-3 -2+3 when simplified is (a) positive and irrational (b) positive and rational (c) negative and irrational (d) negative and rational

-2-3-2+3=-2+3×-2-3=2+32-3 =22-32 a+ba-b=a2-b2=4-3=1 Thus, the given expression when simplified is positive and rational. Hence, the correct answer is option (b).

$(- 2 - \sqrt{3}) (- 2 + \sqrt{3})$ when simplified is
(a) positive and irrational
(b) positive and rational
(c) negative and irrational
(d) negative and rational

$(- 2 - \sqrt{3}) (- 2 + \sqrt{3}) = - (2 + \sqrt{3}) \times [- (2 - \sqrt{3})] = (2 + \sqrt{3}) (2 - \sqrt{3})$
$= 2^{2} - {(\sqrt{3})}^{2} [(a + b) (a - b) = a^{2} - b^{2}] = 4 - 3 = 1$
Thus, the given expression when simplified is positive and rational.

Hence, the correct answer is option (b).