Radical Expressions

Expressions that include roots are known as radical expressions. Recall that the $n$th root of a number $x$ is a number $r$ that, when raised to the power of $n$, equals $x$:

${r}^{n}=x$

where $n$ is the degree of the root. A root of degree 2 is called a square root; a root of degree 3 is called a cube root. Roots of higher degrees are referred to using ordinal numbers (e.g., fourth root, twentieth root, etc.).

Simplified Form

A radical expression is said to be in simplified form if:

1. there is no factor of the radicand that can be written as a power greater than or equal to the index,
2. there are no fractions under the radical sign, and
3. there are no radicals in the denominator.

Example

For example, let's write the radical expression $\sqrt { \frac { 32 }{ 5 } }$ in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign, and remove it:

$\displaystyle \sqrt { \frac { 32 }{ 5 } } =\sqrt { \frac { 16\cdot 2 }{ 5 } } = \sqrt {4^2 \cdot \frac { 2 }{ 5 } }= 4\sqrt { \frac { 2 }{ 5 } }$

Next, separate the fraction under the radical sign:

$\displaystyle 4\sqrt { \frac { 2 }{ 5 } } =\frac { 4\sqrt { 2 } }{ \sqrt { 5 } }$

Finally, remove the radical from the denominator:

$\displaystyle 4\sqrt { \frac { 2 }{ 5 } } =\frac { 4\sqrt { 2 } }{ \sqrt { 5 } } \cdot \frac { \sqrt { 5 } }{ \sqrt { 5 } } =\frac { 4\sqrt { 10 } }{ 5 }$

Radical Expressions with Variables

For the purposes of simplification, radical expressions containing variables are treated no differently from expressions containing integers. For example, consider the following: $$

$\sqrt{4x^2} = \sqrt{4} \cdot \sqrt{x^2} = 2x$

This follows the same logic that we used above, when simplifying the radical expression with integers: 

$\sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$

Example

Simplify the following expression:

$\dfrac{\sqrt{16x^7}}{\sqrt[4]{x^2}}$

First, notice that there is a perfect square under the square root symbol, and pull that out:

$\dfrac{\sqrt{16x^7}}{\sqrt[4]{x^2}} = \dfrac{\sqrt{16}\sqrt{x^7}}{\sqrt[4]{x^2}} = \dfrac{4 \cdot \sqrt{x^7}}{\sqrt[4]{x^2}} = 4 \cdot \dfrac{\sqrt{x^7}}{\sqrt[4]{x^2}}$

Recall that we can rewrite the numerator and denominator in rational exponent form, which will allow us to proceed with the division rule: 

$4 \cdot \dfrac{x^{\frac{7}{2}}}{x^{\frac{2}{4}}}$

Notice that the exponent in the denominator can be simplified, so we have:

$4 \cdot \dfrac{x^{\frac{7}{2}}}{x^{\frac{1}{2}}}$

Recall the rule for dividing numbers with exponents, in which the exponents are subtracted. Applying the division rule yields:

$4 \cdot \dfrac{x^{\frac{7}{2}}}{x^{\frac{1}{2}}} = 4 \cdot x^{\frac{7}{2}-\frac{1}{2}} = 4 \cdot x^{\frac{6}{2}} = 4x^3$