Alternative Format — Lesson 1: Introduction to Radicals

Let's Start Thinking

Radicals

This unit focuses on radicals and rational functions. We will begin by looking at radicals.

You may recall that an irrational number is a real number that cannot be expressed as a fraction of two integers. If written as a decimal, an irrational number would have an infinite number of digits to the right of the decimal point without repetition. Both \(\pi\) and \(\sqrt{2}\) are examples of irrational numbers.

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The square root symbol, \(\sqrt{\quad}\), is an example of a radical symbol. And \(\sqrt{2}\) is an example of a radical. As a result, we might consider that a radical may be used as a compact way of writing some irrational numbers.

Try This

To measure voltage \(V\) (measured in volts), electrical engineers might use the following formula:

where \(P\) is power (measured in watts) and \(R\) is resistance (measured in ohms).

1. What is the approximate voltage if the power is \(1.3\) watts and the resistance is \(5\) ohms?
2. What is the exact voltage if the power is \(450\) watts and the resistance is \(30\) ohms?

A Review of the Number System and the Principal \(n\)^th Root

The Number System

The numbers used thus far in your study of mathematics can be categorized using several terms: natural, whole, integer, rational, irrational, and real.

Here is an overview of the \(6\) number systems:

Natural numbers: \(\{1,~2,~3,~4,~5, \ldots\}\)

The natural numbers are the counting numbers.

The set of natural numbers is denoted by \(\mathbb{N}\).

Some definitions include \(0\) as a natural number; in this courseware, we will use the definition given here.

Whole Numbers: \(\{0,~1,~2,~3,~4, \ldots\}\)

The whole numbers are all the natural numbers together with \(0\).

The set of whole numbers is denoted by \(\mathbb{W}\).

Integers: \(\{\ldots,~{-4},~{-3},~{-2},~{-1},~0,~1,~2,~3,~4, \ldots\}\)

The integers are all the natural numbers and their negatives, as well as \(0\).

The set of integers is denoted by \(\mathbb{Z}\) or \(\mathbb{I}\). The most common notation is \(\mathbb{Z}\).

Rational numbers

The rational numbers are numbers that can be written as fractions of two integers. Any number that is either a terminating or periodic (repeating) decimal can be written as a fraction.

This includes all integers since integers can be written as fractions using a denominator of \(1\).

The set of rational numbers is denoted by \(\mathbb{Q}\).

Irrational numbers

The irrational numbers are numbers that cannot be written as a fraction of two integers. When expressed as a decimal, the decimal is non-terminating and non-repeating.

Examples include \(\pi\) and \(\sqrt{5}\).

The set of irrational numbers is denoted by \(\mathbb{\bar{Q}}\), a symbol that stands for "not \(\mathbb{Q}\)."

Real numbers

The real numbers consist of any number that is either rational or irrational.

The set of real numbers is denoted by \(\mathbb{R}\).

This Venn diagram gives a visual representation of how these number sets relate.

Expand/Collapse Alternative Format for Venn Diagram

Notice that \(\mathbb{R} \) contains all numbers that are in either \(\mathbb{Q}\) or \(\mathbb{\bar{Q}}\).

The Principal \(n\)^th Root

For instance, \(4=2^2\) and \(4=(-2)^2\), therefore \(2\) and \(-2\) are the square roots of \(4\).

Looking at the examples above, although both \(2\) and \(-2\) are the square roots of \(4\), the principal square root of \(4\) is \(2\).

Example 1

Solution

The \(\sqrt{25}\) is asking us to find the positive number whose product with itself is equal to \(25\).

There are two numbers whose product with itself is \(25\).

\(25= 5\times 5\)

or

\(25 = (-5)\times (-5)\)

In a question like this, the word evaluate implies that we are being asked to find the principal square root.  Since \(25\) is positive, we require the positive square root.

\(\begin{align*} \sqrt{25}&=\sqrt{5\times5}\\ &=\sqrt{5^2}\\ &=5 \end{align*}\)

Example 2

Solution

We can rewrite the given question as \( 216^{\frac{1}{3}} =\sqrt[3]{216} \), which means we will need to find a positive number whose cube is \(216\).

Evaluating, we get

\(\begin{align*} 216^{\frac{1}{3}}& \; =\sqrt[3]{216} \\ &\; =\sqrt[3]{6\times6\times6} \\ &\; =\sqrt[3]{6^3} \\ &\; =6 \end{align*}\)

An Introduction to Radicals

Radicals

Mixed and Entire Radicals

There are two different ways that we can represent a radical.

A mixed radical contains an integer part, not \(0\) or \(1\), and a radical part. The operation between \(a\) and \(\sqrt{x}\) is multiplication. You can think of an entire radical as being a mixed radical where \(a=1\). 

How do we know that we have completely simplified a square root? 

Simplifying Radicals

So far, the questions considered have evaluated nicely to whole-number answers like \(\sqrt{25} = 5.\)

You have likely realized that this will not always be possible.

While a radical such as \(\sqrt{18} \) does not evaluate to a whole number, it can be simplified. Before we learn how to do that, we need to recall two important facts.

Important Facts

Let's look at rules for power of a product and power of a quotient. 

Look specifically at the following example with an exponent of \(\dfrac{1}{2}\), which we know can be expressed as a square root.

Since \((xy)^{\frac{1}{2}}=x^{\frac{1}{2}}y^{\frac{1}{2}}\) and \(x^{\frac{1}{2}}=\sqrt{x}\), then \(\sqrt{xy}=\sqrt{x}\times\sqrt{y}\).

Let's return to the example of simplifying \(\sqrt{18} \) to see why the two exponent rules are useful in simplifying radicals.

Example 3

Solution

Approach 1

First, we recall that square root is equivalent to writing the exponent \(\dfrac{1}{2}\).

\( \sqrt{18} = 18^{\frac{1}{2}}\)

Next, we express \(18\) as the product of \(9 \times 2\). Why \(9\) and \(2\)? Why not \(3\) and \(6\)? The answer should become clear as we progress.

\(\begin{align*} \sqrt{18} &= 18^{\frac{1}{2}}\\ &= (9 \times 2)^{\frac{1}{2}} \end{align*}\)

Now we apply the power of a product exponent rule. Then, rewriting the exponent form back as square roots, we get

\(\begin{align*} \sqrt{18} &= 18^{\frac{1}{2}}\\ &= (9 \times 2)^{\frac{1}{2}} \\ &= 9^{\frac{1}{2}} \times 2^{\frac{1}{2}} \\ &= \sqrt{9} \times \sqrt{2} \\ &= 3 \times \sqrt{2} \\ &= 3\sqrt{2} \end{align*}\)

So why did we choose to express \(18\) as \(9 \times 2\) and not, say, \(6 \times 3\)?

The key to simplifying square roots is to find a factor of the radicand, \(18\) in our example, which is a perfect square and take the principal square root of this value. 

Now that we have seen how the exponent rule helps us to simplify the radical, we can streamline our solution as seen in Approach 2. 

Approach 2

\(\begin{align*} \sqrt{18} &= \sqrt{9 \times 2} \\ &= \sqrt{9} \times \sqrt{2} \\ &= 3 \sqrt{2} \end{align*}\)

In this example, \(\sqrt{18}\) is an entire radical and \(3\sqrt{2}\) is a mixed radical.

Example 4

Solution

The word simplify indicates that we need to express this entire radical as a mixed radical.

Approach 1

We could think of \(192\) as \(4 \times 48\). Take the principal square root of \(4\), and we have

\(\begin{align*} \sqrt{192} &= \sqrt{4 \times 48} \\ &= \sqrt{4} \times \sqrt{48} \\ &= 2 \sqrt{48} \end{align*}\)

But we can write \(48\) as \(4 \times 12\), which means we will have 

\(\begin{align*} 2 \sqrt{48} &= 2\sqrt{4 \times 12} \\ &= 2 \times 2 \sqrt{12} \\ &= 4 \sqrt{12} \end{align*}\)

We can keep going, since \(12\) can be written as \(4 \times 3\). Therefore, we have

\(\begin{align*} 4 \sqrt{12} &= 4 \sqrt{4 \times 3} \\ &= 4 \times 2\sqrt{3}\\ &= 8\sqrt{3} \end{align*}\)

Is there a more efficient way to simplify this radical? Yes. We can find the largest perfect square that is a factor of \(192\).

Consider Approach 2, where we start by rewriting \(192\) as \(64 \times 3\).

Approach 2 

Take the principal square root of \(64\) and get

\(\begin{align*} \sqrt{192} &= \sqrt{64 \times 3} \\ &= \sqrt{64} \times \sqrt{3} \\ &= 8\sqrt{3} \end{align*}\)

We can see from these two solutions that it is in our best interests to take a little time to try to find the largest perfect square that is a factor of the radicand.

Example 5

Solution

Just as perfect square factors are key to simplifying square roots, perfect cube factors are key to simplifying cube roots. In this example, we are hoping to find a factor of \(54\), which is a perfect cube.

We might begin by trying \(2^3 = 8\) and recognize that it is not a factor of \(54\).

The next largest perfect cube is \(3^3 = 27\), which is a factor of \(54\).

We proceed by applying our exponent rules as before to separate the principle cube root of \(27\) and the cube root of \(2\). We can simplify cube root \(54\) to 

\(\begin{align*} \sqrt[3]{54} &= \sqrt[3]{27\times2}\\ &=\sqrt[3]{27}\times\sqrt[3]{2} \\ &=3\sqrt[3]{2} \end{align*}\)

We may use this same technique to simplify fourth roots, fifth roots, and any \(n\)^th root.

Check Your Understanding 1

Question

Simplify \(\sqrt{343}\)

Answer

\(7 \sqrt{7}\)

Feedback

\(\begin{align*} \sqrt{343} &= \sqrt{49 \times 7} \\ &= \sqrt{49} \times \sqrt{7} \\ &= 7\sqrt{7} \end{align*}\)

Example 6

Solution

In this example, we need to change the mixed radical to an entire radical. It is sometimes useful to represent radicals as an entire radical, in particular when we are comparing their values, as you'll see in a later example.

To do this, we can take the steps that we have been following to write an entire radical as a mixed radical and use them in reverse.

Begin by writing \(5\) as \(\sqrt{25}\). We then can write \(\sqrt{25 \times 3}\) under the root sign and multiply to get

\(\begin{align*} 5\sqrt{3} &=\sqrt{25}\times\sqrt{3}\\ &=\sqrt{25\times3} \\ &=\sqrt{75} \end{align*}\)

\(5\sqrt{3}\) and \(\sqrt{75}\) are equivalent radicals. They have just been represented in different forms.

Check Your Understanding 2

Question

Express \(6\sqrt{5}\) as an entire radical.

Answer 

\(\sqrt{180}\)

Feedback 

\(\begin{align*} 6\sqrt{5} &= \sqrt{36} \times \sqrt{5} \\ &= \sqrt{36 \times 5} \\ &=\sqrt{180} \end{align*}\)

Example 7

Solution

We recognize that \(\dfrac{2}{3}\) has the same sign as \(\dfrac{4}{9}\) (they are both positive), which is what we are looking for in our principle \(n\)^th root.

Example 8

Solution

Approach 1

Applying our exponent rules, we can write

\(\sqrt{\dfrac{147}{108}}=\dfrac{\sqrt{147}}{\sqrt{108}}\)

So at this point, we're looking to simplify numerator and denominator individually. That is, we first search for a perfect square factor of \(147\) and then \(108\), to obtain

\( \dfrac{\sqrt{147}}{\sqrt{108}}=\dfrac{\sqrt{49\times3}}{\sqrt{36\times3}}\)

This simplifies to

\(\dfrac{7\sqrt{3}}{6\sqrt{3}}\)

After cancelling the \(\sqrt{3}\) from both the numerator and denominator, we're left with

\(\dfrac{7}{6}\)

Approach 2

As is typically the case when working with fractions, reducing them to lowest terms is generally a great first step.

We might have recognized that the numerator \(147\) has a digit sum of \(12\) which is a multiple of \(3\), so \(147\) is divisible by \(3\). Similarly, \(108\) has a digit sum of \(9\) which is a multiple of \(3\), so \(108\) is also divisible by \(3\).

This gives

Check Your Understanding 3

Question

Simplify \(\sqrt{\dfrac{32}{50}}\).

Answer 

\(\dfrac{4}{5}\)

Feedback 

\(\begin{align*} \sqrt{\dfrac{32}{50}} &= \sqrt{\dfrac{16}{25}} \\ &= \dfrac{\sqrt{18}}{\sqrt{25}} \\ &=\dfrac{4}{5} \end{align*}\)

Example 9

Solution

Consider factors of \(30\):

\(2\), \(3\), \(5\), \(6\), \(10\), \(15\), \(30\)

Notice the list of factors does not include any of the perfect squares up to \(30\): \(4\), \(9\), \(16\), \(25\).

Therefore, it is not possible to express \(30\) as the product of a perfect square and some other number, and so \(\sqrt{30}\) is in simplified form.

Check Your Understanding 4

Question

Are the following radicals simplified? Answer Yes or No.

1. \(\sqrt{25}\)
2. \(\sqrt{42}\)
3. \(\sqrt[3]{16}\)
4. \(3\sqrt{5}\)

Answer 

1. No
2. Yes
3. No
4. Yes

Feedback 

1. This is not simplified since \(\sqrt{25}=5\).
2. This is simplified since there are no perfect square factors of \(42\).
3. This is not simplified since \(\sqrt[3]{16} = \sqrt[3]{8 \times 2} = \sqrt[3]{8} \times \sqrt[3]{2} = 2\sqrt[3]{2}\).
4. This is simplified since there are no perfect square factors of \(5\).

Exact and Approximate Values of Radicals

 Example 10

Solution

Unlike the previous example, we do not need to find an exact value. The words "approximate value" give us permission to use technology.

Using the square root button on a calculator, we get

rounded to \(2\) decimal places.

We could also approximate \(\sqrt[5]{30}\).

To do this, we would use the \(n\)^th root button on our calculator (or \(x\)^th root button on some calculators).

Source: Calculator - elsen029/iStock/Getty Images

Or alternatively, you might recall that \(\sqrt[5]{30}\) can be written as \(30^{\frac{1}{5}}\) and use the exponent button to evaluate.

Now try this, what is the approximate value of \(\sqrt[5]{30}\) rounded to the nearest hundredth?  

\(\sqrt[5]{30}\approx 1.97\)

Check Your Understanding 5

Question — Version 1

Determine an approximate value of \(\sqrt[4]{26}\) rounded to the nearest hundredth.

Answer — Version 1

\(2.26\)

Feedback — Version 1

Using the \(n\)^th root button on a calculator, we get 

\(\sqrt[4]{26} \approx 2.26\)

Alternatively, you may use the exponent button and evaluate to get

\(26^{\frac{1}{4}} \approx 2.26\)

Question — Version 2

Determine an approximate value of \(\sqrt{37}\) rounded to the nearest hundredth.

Answer — Version 2

\(6.08\)

Feedback — Version 2

Using the square root button on a calculator, we get 

\(\sqrt{37} \approx 6.08\)

Alternatively, you may use the exponent button and evaluate to get

\(37^{\frac{1}{2}} \approx 6.08\)

Example 11

Solution

Check Your Understanding 6

Question

How do the following radicals compare? Fill in the blank with either \(\gt\) or \(\leq\) sign.

1. \(10\sqrt{6}\)   \(3 \sqrt{10}\)
2. \(4\sqrt{5}\)   \(12\sqrt{2}\)
3. \(7\sqrt{7}\)   \(9 \sqrt{2}\)

Answer

1. \(10\sqrt{6} \gt 3 \sqrt{10}\)
2. \(4\sqrt{5} \leq 12\sqrt{2}\)
3. \(7\sqrt{7} \gt 9 \sqrt{2}\)

Feedback

1. \(10\sqrt{6} = \sqrt{100} \times \sqrt{6} = \sqrt{100 \times 6} = \sqrt{600}\)
   \(3\sqrt{10} = \sqrt{9} \times \sqrt{10} = \sqrt{9 \times 10} = \sqrt{90}\)
   Therefore, \(10\sqrt{6} \gt 3 \sqrt{10}\).
2. \(4\sqrt{5} = \sqrt{16} \times \sqrt{5} = \sqrt{16 \times 5} = \sqrt{80}\)
   \(12\sqrt{2} = \sqrt{144} \times \sqrt{2} = \sqrt{144 \times 2} = \sqrt{288}\)
   Therefore, \(4\sqrt{5} \leq 12\sqrt{2}\).
3. \(7\sqrt{7} = \sqrt{49} \times \sqrt{7} = \sqrt{49 \times 7} = \sqrt{343}\)
   \(9 \sqrt{2}= \sqrt{81} \times \sqrt{2} = \sqrt{81 \times 2} = \sqrt{162}\)
   Therefore, \(7\sqrt{7} \gt 9 \sqrt{2}\).

Interactive Version

Comparing Radicals

Try This Problem Revisited

Solution — Part 1

Substituting the known values, we have 

\(\begin{align*} V&=\sqrt{PR} \\ & =\sqrt{1.3\times5} \\ &=\sqrt{6.5} \end{align*}\)

This question asks us to approximate the voltage. Therefore, we may use technology to find \(V\) which is \(\sqrt{6.5}\approx2.55\) rounded to the nearest hundredth.

So, when the power is \(1.3\) watts and the resistance is \(5\) ohms, the voltage is approximately \(2.55\) volts.

Solution — Part 2

Substituting the known values, we have

\(\begin{align*}V&=\sqrt{PR} \\ &=\sqrt{450\times30} \end{align*} \)

We could multiply these two numbers (\(450\) and \(30\)) to get \(13\;500\), but it's easier for us to simplify this radical by recognizing that \(450\) has a factor of \(30\). Thus, we get

\(\begin{align*} V& =\sqrt{15\times30\times 30} \\ &= \sqrt{15 \times 30^2} \\ & = \sqrt{15} \times \sqrt{30^2} \\ & = 30\sqrt{15} \end{align*}\)

At this point, we confirm that \(15\) has no factors that are perfect squares, so we have completely simplified the radical.

Therefore, when the power is \(450\) watts and the resistance is \(30\) ohms, the exact voltage is \(30\sqrt{15}\) volts.

It is worth noting here, that while \(\sqrt{13\;500}\) is an exact voltage, it is generally understood that we simplify radicals in the same way that we would simplify fractions: without even being asked. Now this question did ask us to simplify our answer, but even if it didn't we would want to write our final answer as \(30\sqrt{15}\).

Wrap-Up

Take It With You

Simplify \(4\sqrt{18}-\sqrt{8}+8\sqrt{2}\).