Question

The conjugate of the surd \[\sqrt 5 - 2\] ?
A. $\sqrt 5 + 2$
B. $ - \sqrt 5 + 2$
C. Either (a) or (b)
D. None of these

Answer 1

Hint: Two quadratic surds can be written as a sum and also as a difference. So these sum and difference simple quadratic surds are conjugate to each other. A surd is an expression that includes a root whether it be a square root, a cube root, or any other root. They are used to represent irrational numbers.

Complete step-by-step solution:
Numbers that cannot be expressed as fractions or recurring decimals are known as surds. They are irrational numbers containing a root, which can be either a square root or a cube root.
A conjugate of a number is the pair of that number joined by different signs. It is formed by changing the sign between two terms or surds.
Two quadratic surds can be written as a sum and also as a difference. So these sum and difference simple quadratic surds are conjugate to each other.
$\sqrt 5 - 2$ is a surd.
According to the definition, the sum and difference of two simple quadratic surds are conjugate to each other.
This means the difference $\left( {\sqrt 5 - 2} \right)$ and the sum $\left( {\sqrt 5 + 2} \right)$ are conjugate to each other.
This implies that $\left( {\sqrt 5 + 2} \right)$ is the conjugate of $\left( {\sqrt 5 - 2} \right)$ .
Therefore, option (a) is correct.
Option (b) is not correct because it changes both the signs which is not the principle of conjugate surds. Therefore, it is incorrect.

Note: The rationalization surds of a number are derived by multiplying the numerator and denominator with the conjugate surd of the denominator. This must be kept in mind that there must be a fractional surd to rationalize and not a simple surd, like the above example.
If the product of two binomial surds is rational then the surds are known as conjugate surds.