Roots are the inverse operation for exponents. An expression with roots is called a radical expression. It's easy, although perhaps tedious, to compute exponents given a root. For instance $7\cdot7\cdot7\cdot7 = 49\cdot49 = 2401$. So, we know the fourth root of 2401 is 7, and the square root of 2401 is 49. What is the third root of 2401? Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division.

Let's go through some basic mathematical operations with radicals and exponents.

Adding and Subtracting Radical Expressions

To add radicals, the radicand (the number that is under the radical) must be the same for each radical, so, a generic equation will have the form:

$a\sqrt{b}+c\sqrt{b} = (a+c)\sqrt{b}$

Let's plug some numbers in place of the variables:

$\sqrt 3 +2\sqrt 3 = 3\sqrt 3$

Subtraction follows the same rules as addition:

$a\sqrt b - c\sqrt b = (a-c)\sqrt b$

For example:

$3\sqrt 3 -2\sqrt 3 = \sqrt 3$

Multiplying Radical Expressions

Multiplication of radicals simply requires that we multiply the variable under the radical signs.

$\sqrt a \cdot \sqrt b = \sqrt {a\cdot b}$

Some examples with real numbers:

$\sqrt 3 \cdot \sqrt 6 = \sqrt {18}$

This equation can actually be simplified further; we will go over simplification in another section.

Simplifying Radical Expressions

A radical expression can be simplified if:

1. the value under the radical sign can be written as an exponent,
2. there are fractions under the radical sign,
3. there is a radical expression in the denominator.

For example, the radical expression $\displaystyle \sqrt{\frac{16}{3}}$ can be simplified by first removing the squared value from the numerator.

$\displaystyle \sqrt{\frac{16}{3}} = \sqrt{\frac{4^2}{3}} = 4\sqrt{\frac{1}{3}}$

Then, the fraction under the radical sign can be addressed, and the radical in the numerator can again be simplified.

$\displaystyle 4\sqrt{\frac{1}{3}} = \frac{4\sqrt{1}}{\sqrt{3}} = \frac{4}{\sqrt{3}}$

Finally, the radical needs to be removed from the denominator.

$\displaystyle \frac{4}{\sqrt{3}} = \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3} = \frac{4}{3}\sqrt{3}$