3.1 Simplify Expressions with Roots

Simplify Expressions with Roots

In Foundations, we briefly looked at square roots. Remember that when a real number [latex]n[/latex] is multiplied by itself, we write [latex]{n}^{2}[/latex] and read it ‘[latex]n[/latex] squared’. This number is called the square of [latex]n[/latex], and [latex]n[/latex] is called the square root. For example,

[latex]\begin{array}{c}{13}^{2}\text{ is read “13 squared”}\\ \text{169 is called the square of 13, since }{13}^{2}=169\\ \text{13 is a square root of 169}\end{array}[/latex]

Can we simplify [latex]\sqrt{-49}[/latex]? Is there a number whose square is [latex]-49[/latex]?

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to [latex]\sqrt{-49}[/latex]. The square root of a negative number is not a real number.

Let’s review some vocabulary first.

[latex]\begin{array}{cccccc}\text{We write:}& & & & & \text{We say:}\\ {n}^{2}& & & & & n\text{ squared}\\ {n}^{3}& & & & & n\text{ cubed}\\ {n}^{4}& & & & & n\text{ to the fourth power}\\ {n}^{5}& & & & & n\text{ to the fifth power}\end{array}[/latex]

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from [latex]−5[/latex] to [latex]5[/latex].

Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of [latex]−2[/latex] to help you see this.

We will now extend the square root definition to higher roots.

[latex]n^{th}[/latex] Root of a Number

[latex]\begin{array}{c}\text{If }b^n=a\text{ , then }b\text{ is an }n^{th}\text{ root of }a\text{.}\\\text{The principal }n^{th}\text{ root of }a\text{ is written as }\sqrt[n]a\text{.}\\n\text{ is called the }\textbf{index}\text{ of the radical.}\end{array}[/latex]

Just like we use the word ‘cubed’ for [latex]b^3[/latex], we use the term ‘cube root’ for [latex]\sqrt[3]{a}[/latex].

We can refer to Table 3.1.1 to help find higher roots.

[latex]\begin{array}{ccc}{4}^{3}& =& 64\\ {3}^{4}& =& 81\\ {\left(-2\right)}^{5}& =& -32\end{array}\phantom{\rule{6em}{0ex}}\begin{array}{ccc}\sqrt[3]{64}& =& 4\\ \sqrt[4]{81}& =& 3\\ \sqrt[5]{-32}& =& -2\end{array}[/latex]

Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of [latex]\sqrt[n]{a}[/latex]

We will apply these properties in the next two examples.

In this example be alert for the negative signs as well as even and odd powers.

Estimate and Approximate Roots

When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the [latex]\sqrt{4}=2[/latex], what is the value of [latex]\sqrt[]{21}[/latex] or [latex]\sqrt[3]{50}[/latex]? In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.

To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of [latex]\sqrt{11}[/latex], we see [latex]11[/latex] is between perfect square numbers [latex]9[/latex] and [latex]16[/latex], closer to [latex]9[/latex]. Its square root then will be between [latex]3[/latex] and [latex]4[/latex], but closer to [latex]3[/latex].

Similarly, to estimate [latex]\sqrt[3]{91}[/latex], we see [latex]91[/latex] is between perfect cube numbers [latex]64[/latex] and [latex]125[/latex]. The cube root then will be between [latex]4[/latex] and [latex]5[/latex].

Example 3.1.5

Estimate each root between two consecutive whole numbers:

a. [latex]\sqrt{105}[/latex]
b. [latex]\sqrt[3]{43}[/latex]

Solution

a.

Step 1: Think of the perfect square numbers closest to [latex]105[/latex].

Make a small table of these perfect squares and their squares roots.

[latex]\sqrt{105}[/latex]

Step 2: Locate [latex]105[/latex] between two consecutive perfect squares.

[latex]{100<}{\color{red}{105}}{<121}[/latex]

[latex]\sqrt{105}[/latex] is between their square roots.

[latex]{10<}{\color{red}{\sqrt{105}}}{<11}[/latex]

---

b.

Step 1: Similarly we locate [latex]43[/latex] between two perfect cube numbers.

[latex]\sqrt[3]{43}[/latex]

Step 2: Locate [latex]43[/latex] between two consecutive perfect cubes.

[latex]{27<}{\color{red}{43}}{<64}[/latex]

[latex]\sqrt[3]{43}[/latex] is between their cube roots.

[latex]{3<}{\color{red}{\sqrt[3]{43}}}{<4}[/latex]

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the [latex]\sqrt{x}[/latex] key on your calculator. To find a cube root, or any root with higher index, you will use the [latex]\sqrt[y]{x}[/latex] key.

When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is [latex]\approx[/latex] and it is read ‘approximately’.

Suppose your calculator has a 10 digit display. You would see that

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

Their squares are close to [latex]5[/latex], but are not exactly equal to [latex]5[/latex]. The fourth powers are close to [latex]93[/latex], but not equal to [latex]93[/latex].

Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,

[latex]\begin{array}{ccccccc}&&\sqrt[3]{{\color{red}{4}}^3}&\text{and}&\sqrt[3]{{({\color{red}{-4}})}^3}&&\\{\color{red}{\text{same}}}&\begin{array}{c}{\color{red}{\nearrow}}\\{\color{red}{\searrow}}\end{array}&\sqrt[3]{64}&&\sqrt[3]{-64}&\begin{array}{c}{\color{red}{\nwarrow}}\\{\color{red}{\swarrow}}\end{array}&{\color{red}{\text{same}}}\\&&{\color{red}{4}}&&{\color{red}{-4}}&&\end{array}\\\text{In either case, when}\;n\;\text{is odd,}\;\sqrt[n]{{\color{red}{a}}^n}={\color{red}{a}}\text{.}[/latex]

But what about an even root? We want the principal root, so [latex]\sqrt[4]{625}=5[/latex].

But notice,

[latex]\begin{array}{ccccccc}&&\sqrt[4]{{\color{red}{5}}^4}&\text{and}&\sqrt[4]{{({\color{red}{-}}{\color{red}{5}})}^4}&&\\{\color{red}{\text{same}}}&\begin{array}{c}{\color{red}{\nearrow}}\\{\color{red}{\searrow}}\end{array}&\sqrt[4]{625}&&\sqrt[4]{625}&\begin{array}{c}{\color{red}{\nwarrow}}\\{\color{red}{\swarrow}}\end{array}&{\color{red}{\text{different}}}\\&&{\color{red}{5}}&&{\color{red}{5}}&&\end{array}\\\text{In either case, when}\;n\;\text{is even,}\;\sqrt[n]{{\color{red}{a}}^n}\neq{\color{red}{a}}\text{.}[/latex]

How can we make sure the fourth root of [latex]−5[/latex] raised to the fourth power is [latex]5[/latex]? We can use the absolute value. [latex]|-5|=5[/latex]. So we say that when [latex]n[/latex] is even [latex]\sqrt[n]{{a}^{n}}=|a|[/latex]. This guarantees the principal root is positive.

Simplifying Odd and Even Roots

Looking now at the square root,

[latex]\begin{array}{ccc}&&\sqrt{a^{2m}}\\\text{Since}\;\left(a^m\right)^2=a^{2m}&&\sqrt{\left(a^m\right)^2}\\\text{Since}\;n\;\text{is even}\;\sqrt[n]{a^n}=\vert a\vert&&\vert a^m\vert\\&&\text{So}\;\sqrt{a^{2m}}=\vert a^m\vert\end{array}[/latex]

We apply this concept in the next example.

The next example uses the same idea for higher roots.

In the next example, we now have a coefficient in front of the variable. The concept [latex]\sqrt{{a}^{2m}}=|{a}^{m}|[/latex] works in much the same way.

But notice [latex]\sqrt{25{u}^{8}}=5{u}^{4}[/latex] and no absolute value sign is needed as [latex]u^4[/latex] is always positive.

This example just takes the idea farther as it has roots of higher index.

The next examples have two variables.