Algebra 1 - Simplifying Radical Expressions

Introduction

Recall the Basic Laws of Radicals:

1. $\sqrt[n]{a^n}=a or {\left(\sqrt[n]{a}\right)}^n=\left|a\right| where a>0 if n is even$.

If $n$ is odd, then $a\in \mathrm{ℝ}$ and $\sqrt[n]{a^n}=a$.

2. $\sqrt[n]{ab}=\sqrt[n]{a}·\sqrt[n]{b} where a>0 and b>0 if n is even$

If $n$ is odd, then $a, b\in \mathrm{ℝ}$

3. $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[b]{b}} where a>0 and b>0 if n is even.$

If $n$ is odd, then $a, b\in \mathrm{ℝ}$ but $b\ne 0$.

4. $\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$

Rules in Simplifying Radicals:

• Express the radicand in factored form. That is, write it as a product of the greatest perfect nth factor and another factor.
• If the radical is a perfect square, factor the radicand, so the largest perfect square factor within the radicand is one of the factors. If the radical is a cube root, factor the radicand, so the largest perfect cube factor within the radicand is one of the factors, and so on.
• Apply the multiplication law for radicals to show the expression as a product of two radical expressions.
• Evaluate and simplify the radical expression completely. 

A simplified radical expression can be noticed if the following conditions are satisfied:

• The radical expression has no perfect square factors or perfect nth factors aside from 1. A radical expression is left with irrational forms if the given radical expression is not a perfect nth root.
• The index of the radical is expressed in its lowest possible form.
• Fractions are no longer observed in the radicand. 
• By rationalizing the denominator, radicals are no longer evident in the denominator.

Important Key Points

For any natural number $n>1$ and any real number $a$, we have: $\sqrt[n]{a}=a^{\frac{1}{n}}$.

Hence, $\sqrt[n]{a}$ represents the nth root of $a.$

To change ${\left(a+b\right)}^{\frac{2}{3}}$ in radical form, we have $\sqrt[3]{{\left(a+b\right)}^2}$.

In a radical expression $\sqrt[n]{x}$, we have the following conditions:

• If $x>0$ and $n$ is even, then we have a positive root (principal root) and a negative root.
• If $x>0$ and $n$ is odd, then we have a positive root. 
• If $x<0$ and $n$ is even, then there are no real roots.
• If $x<0$ and $n$ is odd, there is one negative root.
• If $x<0$ and $n is odd or even,$ then the root is equal to 0.

Steps in Solving Radical Equations:

• Separate the radical expression on one side of the equation. Then, for cases of having a radical expression on both sides, raise both sides of the equation to the same positive integral power. 
• After removing the radical symbols, solve the equation. Repeat the procedures if there are still radicals until you obtain an equation without radicals.
• Solve the resulting equation and verify the obtained roots in the given equation.

Solved Examples

Example 1. What is the simplified form of $\sqrt{243x^8{y}^3{z}^4}$?

Solution:

$\sqrt{81·3x^8{y}^2·y·z^4}=9x^{4}y{z}^2 \sqrt{3y}$

 

Example 2. What is the simplified form of $\sqrt[3]{135x^6{y}^4{z}^5}$?

Solution:

$\sqrt[3]{27\times 5x^6\times y^3\times y\times z^3\times z^2}=3x^{2}yz \sqrt[3]{5y{z}^2}$

 

Example 3. Simplify the radical expression $\sqrt{\frac{48a^8{b}^6}{2a^3{b}^4}}$.

Solution:

$\sqrt{\frac{48a^8{b}^6}{2a^3{b}^4}}=\sqrt{24a^5{b}^2}=\sqrt{4\times 6a^4\times a\times b^2}=2a^{2}b \sqrt{6a}$

 

Example 4. Simplify the radical $\sqrt{\sqrt[3]{x^{18}}}$.

Solution:

$\sqrt{\sqrt[3]{x^{18}}}=\sqrt[2·3]{x^{18}}=\sqrt[6]{x^{18}}=x^{\frac{18}{6}}=x^3$

 

Example 5. FInd the solution(s) of the equation $\sqrt{4x-5}=\sqrt{x+10}$.

Solution:

$\sqrt{4x-5}=\sqrt{x+10}$

${\left(\sqrt{4x-5}\right)}^2={\left(\sqrt{x+10}\right)}^2$

$4x-5=x+10$

$x=5$

Rationalizing the Denominator

Steps in Rationalizing the Denominator:

• If the denominator is a monomial, multiply the numerator and denominator by an expression with factors that will make the exponents of the factors in the radicand of the denominator exactly divisible by the index.
• If the denominator is a binomial, multiply the numerator and denominator by the conjugate of the denominator. 

 

Example 1. What is the simplified form of $\frac{14}{\sqrt{9x^3}}$?

Solution: 

$\frac{14}{\sqrt{9x^3}}=\frac{14}{\sqrt{9x^2·x}}=\frac{14}{3x\sqrt{x}}\to \frac{14}{3x\sqrt{x}}·\frac{\sqrt{x}}{\sqrt{x}}=\frac{14\sqrt{x}}{3x^2}$

 

Example 2. Simplify the radical $\frac{2}{\sqrt{x}+\sqrt{y}}$.

Solution: 

The conjugate of $\sqrt{x}+\sqrt{y}$ is $\sqrt{x}-\sqrt{y}$.

$\frac{2}{\sqrt{x}+\sqrt{y}}·\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}=\frac{2\sqrt{x}-2\sqrt{y}}{x-y}$

Cheat Sheet

• In the expression $\sqrt{2^x}$ where x is a positive integer, the value of x must be any number divisible by the index 2 to generate an expression without radical. However, if all values of x that are not divisible by two are taken, these form an expression with radical.
• The greatest perfect square factor of $1372x^9$ is $196x^8$, and the other factor is $7x$.
• The largest perfect cube factor of $1080a{b}^4$ is $216b^3$, and the other factor is $5ab$.
• A finite nested radical in the form $\sqrt{a\pm 2\sqrt{b}}$ can be simplified to $\sqrt{a}\pm \sqrt{b}$ using the formula $\sqrt{a\pm 2\sqrt{b}}=\sqrt{a+b+2\sqrt{ab}}$ where $a>b$.
• For infinite nested radicals $\sqrt{a+\sqrt{a+\sqrt{a+\sqrt{a+...}}}}$, we may use the shortcut method applying the formula $x=\frac{1}{2}\left(1\pm \sqrt{4n+1}\right)$ where $n$ is the radicand and the sign $\pm$ depends on the operation indicated in the radicand.

Blunder Areas

• The radical expression $\sqrt{\frac{7}{6}}$ is not yet in simplified form because a simplified radical must not contain a fractional radicand.
• The radical expression $\frac{6}{\sqrt{7}}$ is not yet in simplified form since it has a radical in the denominator.
• Not all obtained roots when solving radical equations satisfy the original equations. These roots/solutions are called extraneous roots.
• Laws of integral exponents are also applicable in expressions involving fractional exponents.