1.2 Learn Basic Properties and Operations

This lesson will help you learn the basic properties and rules that govern math problem-solving.

Radicals

In algebra, answers should be written in simplest form. We can simplify a square root by looking for perfect square factors of the number inside the bracket.

Example

Simplify $$\sqrt {75} $$.

$$\sqrt {75} = \sqrt {25} \,\cdot \,\sqrt 3 $$ The first step is to write factors for 75 using at least one perfect square. 25 is a perfect square because $${5^2} = 25$$.
$$\sqrt {25} \,\cdot \,\sqrt 3 = 5\sqrt 3 $$ Find the square root of 25 and write this number in front of the remaining factor $$\sqrt 3$$.

AnswerIn simplest form, $$\sqrt {75}  = 5\sqrt 3 $$.

Sometimes you can factor a number with more than one perfect square factor.

Example

Simplify $$\sqrt {72} $$.

$$\sqrt {72} = \sqrt 9 \,\cdot \,\sqrt 4 \,\cdot \,\sqrt 2 $$ If you didn’t notice that 36 is a factor of 72, you can factor 72 using perfect squares 9 and 4.
$$\begin{array}{c}\sqrt 9 \,\cdot \,\sqrt 4 \,\cdot \,\sqrt 2 \\ = \,\,3\,\cdot \,2\,\cdot \,\sqrt 2 \\ = 6\sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}$$ Multiply the square roots of 9 and 4. Write the product 6 in front of $$\sqrt 2 $$ to show that $$\sqrt 2 $$ is multiplied by 6.

AnswerIn simplest form, $$\sqrt {72} = 6\sqrt 2 $$.