## Numbers and Operations

#### Contents

The square root of a number is a number that is multiplied by itself to give that number. For example, the square root of 9 is 3 because:
$3\cdot3=9$
The $n$th root of a number is multiplied by itself $n$ times to get that number. For example, the fourth root of 81 is 3 because:
$3\cdot3\cdot3\cdot3=81$
A radical symbol $\sqrt{\phantom0}$ is used to represent $n$th roots, where $n$ is a natural number greater than or equal to 2. In the radical $\sqrt[n]{x}$, the index is $n$, which represents the $n$th root of $x$. For example, the index of $\sqrt[3]{x}$ is 3. In the radical $\sqrt{x}$, the index is not shown, but it is understood to be 2. The radicand is the expression under the radical.
• In the radical $\sqrt{x+2}$, the index is 2, and the radicand is $x+2$. It is read as "the square root of the quantity $x$ plus 2."
• In the radical $\sqrt[3]{x^2}$ , the index is 3, and the radicand is $x^2$. It is read as "the cube root of $x$ squared."
A radical with index $n$ is the inverse, or opposite operation, of the exponent $n$. If $n$ is even and $x\geq0$ or if $n$ is odd:
$\sqrt[n]{x^n}=(\sqrt[n]{x})^n=x$
If $n$ is even and $x<0$, then $\sqrt[x]{x}$ is not a real number. For example, $\sqrt[3]{(-4)^3}=-4$, and $\sqrt{-4}$ is not a real number.

The rules for multiplying or dividing radicals are the same as with exponents. A product or quotient under a radical can be split into two separate radicals, and radicals with the same index that are multiplied or divided can be combined into one.

Properties of radicals are used to simplify radical expressions. It is also possible to combine like radicals, which have the same radicand and index.

Property Example Description
$\sqrt[n]{xy}=\sqrt[n]{x}\sqrt[n]{y}$
$\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}$

Step-By-Step Example
Simplify the expression.
$\sqrt{2}\sqrt{3}+3\sqrt{6}$
Step 1
Use the product of radicals property to combine $\sqrt{2}$ and $\sqrt{3}$.
$\begin{gathered}\sqrt{2}\sqrt{3}+3\sqrt{6}\\\sqrt{2\cdot3}+3\sqrt{6}\\\end{gathered}$
Step 2
Simplify the expression. Multiply 2 and 3 inside the radical.
$\begin{gathered}\sqrt{2\cdot3}+3\sqrt{6}\\\sqrt{6}+3\sqrt{6}\end{gathered}$
Solution
Combine like radicals. The terms $\sqrt{6}$ and $3\sqrt{6}$ have the same radicand and index. So, they can be combined like other like terms by adding the coefficients. The coefficient of $\sqrt{6}$ is 1.
$\begin{gathered}\sqrt{6}+3\sqrt{6}\\4\sqrt{6}\end{gathered}$

A radical expression in simplest form has no perfect powers or fractions in the radicand and no fractions in the denominator of a fraction.
When simplifying radicals, there is a set of rules that determines whether the radical is in simplest form.

Rule Example
There are no perfect $n$th powers under a radical of index $n$, such as perfect squares within a square root or perfect cubes under a cube root.
\begin{aligned}\sqrt{8}&=\sqrt{(4)(2)}\\&=\sqrt{4}\sqrt{2}\\&=2\sqrt{2}\end{aligned}
There are no fractions under a radical or radicals in the denominator of a fraction.
\begin{aligned}\sqrt{\frac{3}{4}}&=\frac{\sqrt{3}}{\sqrt{4}}\\&=\frac{\sqrt{3}}{2}\end{aligned}

Step-By-Step Example
Simplify $\sqrt[4]{x^7}$ for $x\geq0$.
Step 1
The index of the radical is 4. Rewrite $x^7$ as a product of a perfect fourth power and another power.
$\begin{gathered}\sqrt[4]{x^7}\\\sqrt[4]{x^4x^3}\end{gathered}$
Step 2
Use the product of radicals property to split the radical into a product.
$\begin{gathered}\sqrt[4]{x^4x^3}\\\sqrt[4]{x^4}\sqrt[4]{x^3}\end{gathered}$
Solution
Use the inverse relationship between the radical and exponent to rewrite $\sqrt[4]{x^4}$. This expression is equal to $x$ because $x\geq0$.
$\begin{gathered}\sqrt[4]{x^4}\sqrt[4]{x^3}\\x\sqrt[4]{x^3}\end{gathered}$

### Rationalizing the Denominator

Fractions with radicals in the denominator can be simplified by multiplying by a fraction equivalent to 1.

Rationalizing the denominator is the process of rewriting a fraction to remove radical expressions from the denominator. To rationalize the denominator, multiply the fraction by a fraction that is equivalent to 1. The type of radical expression in the denominator determines which fraction equivalent to 1 should be used.

If the denominator has two terms, such as $\sqrt{x}+y$, the conjugate of the expression is $\sqrt{x}-y$. When two expressions are multiplied, the resulting expression does not contain a radical:
$\begin{gathered}\left(\sqrt{x}+y\right)\left(\sqrt{x}-y\right)\\\left(\sqrt{x}\right)^2-y\sqrt{x}+y\sqrt{x}-y^2\\x-y^2\end{gathered}$

### How to Rationalize Denominators

Denominator Fraction Equivalent to 1 Example
Single square root Multiply by a fraction in this form:
$\frac{\sqrt{x}}{\sqrt{x}}=1$
\begin{aligned}\frac{2}{\sqrt{5}}\left(\frac{\sqrt{5}}{\sqrt{5}}\right)&=\frac{2\sqrt{5}}{\sqrt{25}} \\&= \frac{2\sqrt{5}}{5}\end{aligned}
Single cube root Multiply by a fraction in this form:
$\frac{\sqrt[3]{x^2}}{\sqrt[3]{x^2}}=1$
\begin{aligned}\frac{2}{\sqrt[3]{5}}\left(\frac{\sqrt[3]{5^2}}{\sqrt[3]{5^2}}\right)&=\frac{2\sqrt[3]{5^2}}{\sqrt[3]{5^3}}\\&=\frac{2\sqrt[3]{5^2}}{5}\end{aligned}
Expression with two terms, such as:
$\sqrt{x}+y$
Multiply by a fraction in this form:
$\frac{\sqrt{x}-y}{\sqrt{x}-y}=1$
\begin{aligned}\frac{2}{\sqrt{5}+1}\left(\frac{\sqrt{5}-1}{\sqrt{5}-1}\right)&=\frac{2\sqrt{5}-2}{\sqrt{25}-\sqrt{5}+\sqrt{5}-1}\\&=\frac{2\sqrt{5}-2}{4}\end{aligned}

Rational exponents can be used to represent radicals, or $n$th roots.
Radicals can be expressed by using rational exponents. A rational exponent is an exponent that is a rational number. If the exponent is written as a fraction $\frac{a}{b}$, then the denominator $b$ is the index of the radical, and the numerator $a$ is the exponent of the radicand. Alternately, the expression can be written in two ways, with the entire radical raised to the $a$ power:
$x^{\frac{a}{b}}=(x^a)^\frac{1}{b}=\sqrt[b]{x^a}=(\sqrt[b]{x})^a$
$x^{\frac{a}{b}}=(x^\frac{1}{b})^a=\sqrt[b]{x^a}=(\sqrt[b]{x})^a$
$x^{\frac{1}{b}}=\sqrt[b]{x}$
$x^{\frac{1}{2}}=\sqrt{x}$