Algebra
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Boundless Algebra
Introduction to Equations, Inequalities, and Graphing
Variables and Expressions
Algebra Textbooks Boundless Algebra Introduction to Equations, Inequalities, and Graphing Variables and Expressions
Algebra Textbooks Boundless Algebra Introduction to Equations, Inequalities, and Graphing
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 15
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Simplifying Radical Expressions

A radical expression that contains variables can often be simplified to a more basic expression, much as can expressions involving only integers.

Learning Objective

  • Simplify radical expressions containing variables


Key Points

    • If a radical is fully simplified, there is no factor of the radicand that can be written as a power greater than or equal to the index, there are no fractions under the radical sign, and there are no radicals in the denominator.
    • When radical expressions contain variables, simplifying them follows the same process as it does for expressions containing only integers.
    • Similarly, the rules for multiplying and dividing radical expressions still apply when the expressions contain variables.

Term

  • radicand

    The number or expression whose square root or other root is being considered; e.g., the 3 in $\sqrt[n]{3}$. More simply, the number under the root symbol.


Full Text

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:

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

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