Rules to simplify square root expression

In this chapter we will learn to important rules to simply expressions with square roots, cube roots etc.

Some problems are also provided for your practice.

Product rule of roots

Multiplication of two or more root numbers can be done by simply multiplying the numbers inside the root.

\mathtt{\sqrt{a} .\sqrt{b} =\sqrt{ab}}

Note that both numbers “a” & “b” are square root numbers.

Similarly if the numbers are in cube root, the multiplication can be expressed as;

\mathtt{\sqrt[3]{a} .\sqrt[3]{b} =\sqrt[3]{ab}}

Note:
The multiplication is only possible when both numbers have same form of root. You cannot multiply square root number with cube root number.

I hope you understood the concept, let’s solve some problems related to it.

Example 01
Simplify the below expression.
\mathtt{\sqrt{5} \times \sqrt{3}}

Solution
Using the multiplication rule of square roots.

\mathtt{\Longrightarrow \ \sqrt{5} \times \sqrt{3}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{15}}

Hence, the above expression is the solution.

Example 02
Simplify the below expression;
\mathtt{\sqrt[4]{10} \times \sqrt[4]{5} \times \sqrt[4]{7}}

Solution
All the numbers are in root of 4, so multiplication is possible.

\mathtt{\Longrightarrow \ \sqrt[4]{10\times 5\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[4]{350}}

Hence, the above expression is the solution.

Example 03
Simplify the below expression
\mathtt{\sqrt[3]{5} \times \sqrt{6}}

Solution
One number is in form of cube root while other is in form of square root. So multiplication inside the root is not possible.

Division rule of roots

Division of two or more root numbers can be done by simply dividing number inside the root.

\mathtt{\frac{\sqrt{a}}{\sqrt{b}} =\sqrt{\frac{a}{b}}}

Note that both numbers “a” and “b” are in the form of square root.

Similarly if the numbers are in form of cube roots, the division of numbers can be done individually inside the root.

\mathtt{\frac{\sqrt[3]{a}}{\sqrt[3]{b}} =\sqrt[3]{\frac{a}{b}}}

Note:
This division is only possible when both the given numbers are in the form of same root. If one number is in form of square root while other is in form of cube root then division is not possible.

Let’s solve some problem related to the concept.

Example 01
Simplify the expression \mathtt{\frac{\sqrt{10}}{\sqrt{5}}}

Solution
Both the numbers are in form of square root so the division can be done as per above rule.

\mathtt{\Longrightarrow \ \sqrt{\frac{10}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{2}}

Hence, \mathtt{\sqrt{2}} is the right answer.

Example 02
Simplify the expression \mathtt{\frac{\sqrt[3]{125}}{\sqrt[3]{5}}}

Solution
Both the numbers are in form of cube root, so we can apply the above rule.

\mathtt{\Longrightarrow \ \sqrt[3]{\frac{125}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[3]{25}}

Hence, \mathtt{\sqrt[3]{25}} is the solution.

Question 03
Solve the expression \mathtt{\frac{\sqrt[3]{11}}{\sqrt{2}}}

Solution
Here one number is in form of cube root while other is in form of square root.
So, we cannot apply the above division method in this question.

Next chapter : Questions on square root simplification