Question

What is the square root of 6 in simplest radical form?

Answer 1

Hint: To solve the given question we will need the following properties. We should know that the square root of 6 can be algebraically expressed as \[{{\left( 6 \right)}^{\dfrac{1}{2}}}\]. We will need the property of exponents which states that \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]. We will use these properties to solve the given question.

Complete step by step solution:
For the given question, we should know that any term inside the square root can be simplified if it has a factor which is a perfect square. This factor should not be one.
Let’s take an example, say we are asked to simplify the square root of 8. As we know that the factors of 8 are 1, 2, and 4. As 4 is a perfect square, we can simplify it as \[{{\left( 8 \right)}^{\dfrac{1}{2}}}={{\left( 2\times 4 \right)}^{\dfrac{1}{2}}}\]. Using the algebraic property \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\], we get \[{{\left( 2\times 4 \right)}^{\dfrac{1}{2}}}={{\left( 2 \right)}^{\dfrac{1}{2}}}\times {{\left( 4 \right)}^{\dfrac{1}{2}}}=2\sqrt{2}\].
Let’s look at the given example now. We are asked to find the simplest radical form of square root of 6. The factors of 6 are 1, 2, and 3. Excluding one, none of them is a perfect square, so the square root of one can not be further simplified.
It can be written as \[{{\left( 6 \right)}^{\dfrac{1}{2}}}\].

Note: These types of questions can be solved by remembering the values of squares, cubes, square roots, and cube roots of the numbers. If we are asked to write the simplest radical form of cube root, then we need to find a factor which is a perfect cube and not one.