nth roots
Square roots
If you multiply a number by itself, you get a number called its square. Conversely, if you are given a number and you find a number that, multiplied by itself, gives the original number, then that number is called a square root of the given number. So, taking a square root is the reverse operation of squaring.
For example, $5^2=25$, here $25$ is the square of the number $5$.
And for the number $25$, both $5$ and $-5$ are square roots, since multiplying either $5$ or $-5$ by itself gives $25$.
Example: Find the square root of $81$.
Solution:
We can write,
$81=9^2$ and
$81=(-9)^2$
So, the square roots of $81$ are $9$ and $-9$.
Example: Find the square root of $-36$.
Solution:
We cannot find a real number that, multiplied by itself, gives $-36$. Therefore, the square root of $-36$ is not real. So, a negative number has no real square root.
The radical sign
We saw that a positive number has both a positive and a negative square root. For the positive square root, we use the sign $\sqrt{\hphantom{00}}\:$, called the radical sign.
That is, for a positive number $a$, $\:\sqrt{a}$ is the positive square root of $a$.
Examples:
$\sqrt{49}=7$
$\sqrt{36}=6$
And
$\sqrt{0}=0$
Since the square root of a negative number is not a real number,
$\sqrt{-9}=$ not real.
You can have a minus sign outside the radical sign, but not inside it.
That is, $-\sqrt{25}=-5$
Simplifying square roots
Since the square root of a number is just another number, you treat a square root like a number when simplifying. The following relations hold for square roots.
$\sqrt{ab}=\sqrt{a}\sqrt{b}$
$\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$
Example: Simplify $\sqrt{\dfrac{64}{25}}$.
Solution:
$\begin{align*} \sqrt{\dfrac{64}{25}}&=\dfrac{\sqrt{64}}{\sqrt{25}}\\ &=\boxed{\dfrac{8}{5}}\\ \end{align*}$
nth root
Like the square root, we can have a third root, a fourth root, a fifth root, and so on. The third root of a number is a number that, when multiplied three times, equals the original number. The third root is also called the cube root. The fourth root of a number is a number that, when multiplied four times, gives the original number.
If $a$ is a square root of a number, $b$, then $a^2=b$.
If $a$ is a cube root of $b$, then $a^3=b$.
If $a$ is a fourth root of $b$, then $a^4=b$, and so on.
In general, we call these roots nth roots: if $n=2$, it is a square root; if $n=3$, the cube root; if $n=4$, the fourth root; and so on.
So, if $a$ is the $n$th root of a number $b$, then
$a^n=b$.
For a square root, we use the radical sign $\sqrt{\hphantom{0}}\:$; for an nth root, we use $\sqrt[n]{\hphantom{0}}\:$. So for the cube root, we use $\sqrt[3]{\hphantom{0}}\:$; for the fourth root, $\sqrt[4]{\hphantom{0}}\:$; and so on. The number $n$ is the index of the root.
Examples:
$\sqrt[3]{27}=3$
$\sqrt[3]{-64}=-4$
$\sqrt[5]{32}=2$
Note that for even $n$, the $n$th root of a positive number is always positive, and the $n$th root of a negative number is not real. For odd $n$, the $n$th root is positive for a positive number and negative for a negative number.
For $n$ odd, $\sqrt[n]{a^n}=a$
For $n$ even, $\sqrt[n]{a^n}=|a|$.
Example 1: Evaluate, $\sqrt{(-11)^2}$
Solution:
$\begin{align*} \sqrt{(-11)^2}&=|-11|\\ &=11\\ \end{align*}$
Example 2: Evaluate, $\sqrt{(-3)^4}$
Solution:
$\begin{align*} \sqrt{(-3)^4}&=\sqrt{((-3)^2)^2}\\ &=|(-3)^2|\\ &=9\\ \end{align*}$
Example 3: Evaluate, $\sqrt{u^6}$, where $u$ represents a positive real number.
Solution:
$\begin{align*} \sqrt{u^6}&=\sqrt{(u^3)^2}\\ &=|u^3|\\ \end{align*}$
Since $u$ is positive, we can drop the absolute value symbol, and we get
$\sqrt{u^6}=u^3$.
Example 4: Evaluate, $\sqrt{y^{30}}$, where $y$ represents any real number.
Solution:
$\begin{align*} \sqrt{y^{30}}&=\sqrt{(y^{15})^2}\\ &=|y^{15}|\\ \end{align*}$
We are told that $y$ is any real number, so $y$ raised to an odd power is positive when $y$ is positive and negative when $y$ is negative. Since the square root is always positive, we keep the absolute value symbol to ensure the answer is always positive. Therefore,
$\sqrt{y^{30}}=|y^{15}|$.
Simplifying nth roots
Example 1: Simplify, $\sqrt[3]{64}$
Solution:
$\sqrt[3]{64}=\sqrt[3]{4^3}$
$\hphantom{0000}=4$
Example 2: Simplify, $\sqrt[4]{16}$
Solution:
$\sqrt[4]{16}=\sqrt[4]{2^4}$
$\hphantom{0000}=2$
Example 3: Simplify, $\sqrt[3]{\dfrac{64}{27}}$
Solution:
$\begin{align*} \sqrt[3]{\dfrac{64}{27}} &= \dfrac{\sqrt[3]{64}}{\sqrt[3]{27}}\\ &= \dfrac{\sqrt[3]{4^3}}{\sqrt[3]{3^3}}\\ &=\dfrac{4}{3}\\ \end{align*}$
Example 4: Simplify, $\sqrt[5]{243x^{10}}$
Solution:
$\begin{align*} \sqrt[5]{243x^{10}} &= \sqrt[5]{243}\,\sqrt[5]{x^{10}}\\ &= \sqrt[5]{3^5}\,\sqrt[5]{(x^2)^5}\\ &= 3x^2\\ \end{align*}$
Example 5: Simplify, $\sqrt[8]{(13-5u)^8}$, where $u$ represents any real number.
Solution:
Since the index of the radical is even, the root must be positive. Therefore,
$\sqrt[8]{(13-5u)^8}=|13-5u|$.
Radical function
An $n$th root of an expression, written with a radical sign, is called a radical. A function that is equal to a radical is called a radical function.
Examples:
$f(x)=\sqrt{x}$
$g(t)=\sqrt{3t-1}$
$h(x)=\sqrt[3]{2x+5}$
Evaluating a radical function
Evaluating a radical function is the same as evaluating any other function.
Example: Given, $f(x)=\sqrt[3]{2x}-5$. Find $f(-4)$ and $f(32)$.
Solution:
To find $f(-4)$, substitute $x=-4$ into the function $f$:
$f(-4)=\sqrt[3]{2\cdot (-4)}-5$
$\phantom{00000}=\sqrt[3]{-8}-5$
$\phantom{00000}=-2-5$
$\phantom{00000}=-7$
To find $f(32)$, substitute $x=32$ into $f(x)$:
$f(32)=\sqrt[3]{2\cdot 32}-5$
$\phantom{00000}=\sqrt[3]{64}-5$
$\phantom{00000}=4-5$
$\phantom{00000}=-1$
Domain of radical functions
The domain of a radical function $f(x)$ is the set of all $x$ values that do not produce a non-real number.
If the index $n$ of a radical function is odd, then the domain is all real numbers. But if $n$ is even, then the domain is all real numbers that produce a non-negative value under the radical sign.
To find the domain of an even-index radical function, set the expression inside the radical sign greater than or equal to zero, and solve the resulting inequality for the variable. The solution is the domain of the radical function.
Example 1: Find the domain of the function $f(x)=\sqrt[3]{3x-11}$
Solution:
The index of the radical is $3$, which is an odd number; therefore, the domain of the function $f(x)$ is all real numbers.
In interval notation, the domain of $f$ is therefore $(-\infty,\infty)$.
Example 2: Find the domain of the function $h(x)=\sqrt{8x+4}$
Solution:
The index of the radical is $2$ (a square root), which is even, so we cannot have a negative number under the radical sign. Therefore, to find the domain, set the expression under the radical sign greater than or equal to zero and solve for $x$. The solution is the domain of the function $h$.
Setting the expression under the radical sign greater than or equal to zero:
$8x+4\ge 0$
Solving for $x$,
$x\ge -\dfrac{1}{2}$
So, the domain is all $x$ values greater than or equal to $-\dfrac{1}{2}$.
In interval notation, the domain of $h$ is $[-\dfrac{1}{2},\infty)$.
Graphing a radical function
To graph a radical function $f(x)$, exclude the $x$ values that are not in the domain. Pick values of $x$ in the domain and find the corresponding function values (the $y$ values). Then plot the ordered pairs.