Question

How do you simplify \[\sqrt {30} \]?

Answer 1

Hint:Here, we will use the property of surds and the factors of the surds to find the simplified surds. Simplified surd is always a combination of a rational number and an irrational number. Thus, the simplified expression is the required answer.

Formula Used:
Property of Surds: \[\sqrt {a \cdot b} = \sqrt a \cdot \sqrt b \].

Complete Step by Step Solution:
We are given with a mathematical expression \[\sqrt {30} \].
The given mathematical expression is in the form of surds. So, it cannot be simplified since the given surd is not a factor of any perfect square number. So, the surd can be rewritten as
\[ \Rightarrow \sqrt {30} = \sqrt {2 \times 3 \times 5} \].
By using the Property of surds \[\sqrt {a \cdot b} = \sqrt a \cdot \sqrt b \], we get
\[ \Rightarrow \sqrt {30} = \sqrt 2 \cdot \sqrt 3 \cdot \sqrt 5 \].

Therefore, \[\sqrt {30} \] cannot be simplified and \[\sqrt {30} \] can be rewritten as \[\sqrt 2 \cdot \sqrt 3 \cdot \sqrt 5 \].

Additional Information:
We know that surds having the same common surds are called simple surds. We know that the surd made of two surds is defined as the binomial surds. We have to rationalize the denominator which helps in removing the surd from the denominator. Rationalizing the denominator is a method of eliminating the radical expressions in the denominator such as the square roots or cube roots by multiplying with the conjugate to make the denominator a rational number. We can also find the greatest square factor to solve the surds easily.

Note: We know that surds are the numbers that are not the perfect squares. Surds cannot be expressed as a rational number or a whole number. An expression that has the same surds can be added, subtracted, multiplied, or divided and thus the surds are compound surds. Surds that are completely irrational are called pure surds. Surds that are not completely irrational and can be expressed as the product of a rational number and also an irrational number.