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Difficulty Level: At Grade Created by: CK-12

## Introduction

The Food Drive

The eighth grade student council decided that the theme for the school year would be “Helping Hands.” With this theme, the whole year would focus the eighth graders around different community service projects. When the president of the class, Margaret, proposed this to the student body, the students were all very excited. They decided to let each home room figure out what project they were each going to focus on.

Mrs. Garibaldi’s class held a canned-food drive to aid a local relief shelter. Juan was the team leader. He sent out a notice for each family to begin gathering food in the beginning of November. He figured that they could collect all of their cans by Thanksgiving and provide some families with extra food for the holidays.

They collected 121 cans for the shelter. Many different types of canned food was collected. Juan calculated that the number of cans containing vegetables was equal to 12112+14\begin{align*}121^{\frac{1}{2}}+14\end{align*}.

“How many cans contained vegetables?” Margaret asked Juan at lunch.

Juan simply smiled and wrote his expression on a piece of paper.

“That is a weird way to write it!” she said. “I still don’t know how many cans contained vegetables!”

Do you know? This lesson will work with fractional exponents and radicals. At the end of the lesson, you will know how to help Margaret figure things out.

What You Will Learn

In this lesson you will learn how to execute the following skills.

• Evaluate radical expressions involving perfect squares and perfect cubes.
• Approximate square roots and cube roots using rounding, calculators or tables.
• Recognize the equivalence of radical expressions and fractional powers.
• Model and solve real – world problems involving radical expressions.

Teaching Time

I. Evaluate Radical Expressions Involving Perfect Squares and Perfect Cubes

Do you remember exponents? An exponent is a number that raises a base to a power. We can recognize exponents because they are little numbers next to larger numbers. The little number is the exponent and the large number is the base. The exponent tells you how many times to multiply a base by itself.

Look at this example.

Example

72\begin{align*}7^2\end{align*}

This means that we multiply the base of 7 by itself two times. This is how we evaluate a power.

7×7=49\begin{align*}7 \times 7 = 49\end{align*}

We can also perform an operation that is the opposite of raising a number to a power; we can find the root of a number. This is an expression that is the opposite of raising a number to a power. We call it a root or a radical.

When you see a number that looks like this, )16¯¯¯¯¯¯¯\begin{align*}\overset{}{\overline{ ) {16}}}\end{align*}, this means that we are looking for the root of the number that is inside the radical symbol.

Now let’s look at how we can work with roots and radicals.

Example

72=7×7=49\begin{align*}7^2 = 7 \times 7 = 49\end{align*}

If we use verbal language to explain this, we can say that seven squared is equal to 49. When the exponent is a 2, we can say that the number is squared because it is multiplied by itself.

We can work in the opposite of squaring, and find the square root of a number.

Example

)49¯¯¯¯¯¯¯\begin{align*}\overset{}{\overline{ ) {49}}}\end{align*}

When we see a number inside the radical symbol, we are looking to figure out the square root of that number. In other words, what times itself two times is equal to the value inside the radical symbol.

The answer is 7 because 7 squared is equal to 49.

We can also cube a number. When a number is cubed, the exponent is a 3. This means that we multiply the base by itself three times.

Example

23\begin{align*}2^3\end{align*}

This means that we multiply the base two by itself three times.

2×2×2=8\begin{align*}2 \times 2 \times 2 = 8\end{align*}

We can also find the cube root of a number.

Example

83\begin{align*}\sqrt[3]{8}\end{align*}

When looking for a cube root, we are looking for a value that we can multiply by itself three times.

The cube root of 8 is 2.

We can determine if a value is a perfect square or a perfect cube.

Example

Is 64 a perfect square? Is it a perfect cube?

To figure this out, we need to look and see if there is one number that we can multiply by itself to equal 64. This means that we are looking for the square root of 64.

)64¯¯¯¯¯¯¯=8\begin{align*}\overset{}{\overline{ ) {64}}}=8\end{align*}

Since 82=64\begin{align*}8^2 = 64\end{align*}, 64 is a perfect square.

Now is it a perfect cube? To figure this out, we need to see if there is a value that when cubed is equal to 64.

643=4\begin{align*}\sqrt[3]{64}=4\end{align*}

Since 4×4×4=64\begin{align*}4 \times 4 \times 4 = 64\end{align*}, 64 is a perfect cube.

64 is a perfect square and a perfect cube.

Example

Is 40 a perfect square?

To figure this out, we need to figure out the square root of 40.

)40¯¯¯¯¯¯¯\begin{align*}\overset{}{\overline{ ) {40}}}\end{align*}

There is not a number that when squared is equal to 40 without any remainders, so 40 is not a perfect square.

Write down the definition of a radical expression, a square root and a cube root in your notebook. Also be sure that you understand how to figure out if a number is a perfect square or a perfect cube.

II. Approximate Square Roots and Cube Roots Using Rounding, Calculators or Tables

Roots are pretty easy to find when the numbers are perfect squares or perfect cubes. When you can’t find a root that easily squares or cubes, you will need to use a different method. If you just need to estimate a square root, you can identify which two numbers a root would be between.

If you need more information, you can refer to a table that shows square roots, or use a calculator to find an exact decimal point.

That is exactly what we are going to look at next.

Example

Evaluate 30\begin{align*}\sqrt{30}\end{align*}

We know by looking at 30 that it is not a perfect square. Therefore, we will need to approximate the square root. We can do this by figuring out the two values that the square root will be between.

Which values squared will equal a number close to 30?

5262=25=36\begin{align*}5^2 &= 25\\ 6^2 &= 36\end{align*}

We can say that the square root of 30 is between 5 and 6. Since 25 and 36 are almost the same number of units apart, we can say that an approximate answer for the square root of 30 is 5.5.

Example

Evaluate )62¯¯¯¯¯¯¯\begin{align*}\overset{}{\overline{ ) {62}}}\end{align*}

Let’s think about the two values that the square root of 62 will be between because 62 is not a perfect square.

7×7=498×8=64\begin{align*}7 \times 7 = 49\\ 8 \times 8 = 64\end{align*}

We can say that the answer will be between 7 and 8, but very close to 8 because 64 is very close to 62.

Sometimes, you will want an answer that is exact. An approximate answer will not work, when this happens, you will need to use a calculator or a table. There are tables that will tell you the exact square root of a number.

Example

Use your calculator to find the value of 42\begin{align*}\sqrt{42}\end{align*}.

Type the square root symbol and 42 to find the square root of 42 on your calculator.

The result shown is 6.48074069840786. You can round this to the nearest hundredth and record the value as 6.48.

If you wanted to use a table, you can find one in a textbook or on the computer. There are many websites where they will give the square root of numbers from 1 to 100. Here is one such website, http://www.factmonster.com/ipka/A0875883.html.

Now let’s look at an example.

Example

Evaluate )11¯¯¯¯¯¯¯\begin{align*}\overset{}{\overline{ ) {11}}}\end{align*}.

Using the table, we can see that the square root of 11 is 3.317.

III. Recognize the Equivalence of Radical Expressions and Fractional Powers

We began this lesson by looking at statements of equivalence.

Example

72)49¯¯¯¯¯¯¯=49=7\begin{align*}7^2 &= 49\\ \overset{}{\overline{ ) {49}}}&=7\end{align*}

We can also make a connection between radical expressions and fractional powers. A fractional power is when the exponent is in the form of a fraction. Square roots and cube roots can also be represented by fractional exponents. If a number is raised to the power of 13\begin{align*}\frac{1}{3}\end{align*}, it is the same as taking the cube root. Similarly, if a number is raised to the power of 12\begin{align*}\frac{1}{2}\end{align*}, it is the same as taking the square root.

Example

Elena was asked to find the value of 2713\begin{align*}27^{\frac{1}{3}}\end{align*}. How should Elena find this value?

The first step Elena should take is to convert the fractional exponent to a root. Since the fraction is 13\begin{align*}\frac{1}{3}\end{align*}, she will need to find the cube root of 27 to solve the problem.

2713=273\begin{align*}27^{\frac{1}{3}}=\sqrt[3]{27}\end{align*}

To simplify the cube root, Elena should think of a number that, when multiplied three times in a row, yields 27.

If you multiply 3×3×3\begin{align*}3 \times 3 \times 3\end{align*}, the product is 27. So, the cube root of 27 is 3.

2713=273=3\begin{align*}27^{\frac{1}{3}}=\sqrt[3]{27}=3\end{align*}

Example

Birgit needs to solve the equation x=8112\begin{align*}x=81^{\frac{1}{2}}\end{align*}. How can she find the value of x\begin{align*}x\end{align*}?

Birgit should realize that raising something to the power of 12\begin{align*}\frac{1}{2}\end{align*} is the same as taking the square root. So, Birgit simply needs to find the square root of 81 to find the value of x\begin{align*}x\end{align*}.

To find the square root of 81, Birgit could use her calculator, or think about what number, when multiplied by itself, will yield a product of 81.

Since 9×9=81\begin{align*}9 \times 9=81\end{align*}, the square root of 81 is 9.

In the equation x=8112,x=9\begin{align*}x=81^{\frac{1}{2}}, x=9\end{align*}.

We can also see radicals in an equation. When we have a radical in an equation, we can solve the equation by using what we have learned about squares and square roots. Let’s look at an example.

Example

x2=100\begin{align*}x^2=100\end{align*}

To solve this equation let’s think about what we know. We know that a value times itself is going to equal 100. Up until this point we have solved equations by using the inverse operation. We can use that again here too. The variable is squared; we want to get it by itself by using an inverse operation. The inverse of squaring a number is to find the square root of the number. Let’s see what happens if we find the square root of both sides of the equation.

x2=100\begin{align*}\sqrt{x^2}= \sqrt{100}\end{align*}

The exponent 2 and the radical cancel each other out because they are inverses of each other. Let’s cross them out.

x2=100\begin{align*}\sqrt{x^2}=\sqrt{100}\end{align*}

Now are left with:

xx=100=10\begin{align*}x&=\sqrt{100}\\ x&=10\end{align*}

V. Model and Solve Real – World Problems Involving Radical Expressions

We can find radical expressions in real – world problems too. As always, when you face complicated word problems, carefully identify the important information and set up an equation. Remember to translate phrases into mathematical expressions carefully. Always use common sense when figuring out complicated situations. Once you have an equation, use inverse operations to solve for the variable.

Example

Mario and his brother have built a garden bed with an area of 144 square feet. If the shape of the garden is a square, then what is the length of one side of the garden? What is the perimeter of this garden?

First, we have to break down what we know about the garden. We know the area.

A=144 sq.feet\begin{align*}A = 144 \ sq.feet\end{align*}

We also know that the formula for area is s2\begin{align*}s^2\end{align*}.

Now we can write an equation to figure out the first part of the problem, the length of one of the sides.

s2=144 sq.feet\begin{align*}s^2=144 \ sq.feet\end{align*}

If we take the square root of both sides, then we will be able to solve for the value of the side of the garden.

sqrts2s=144=12 feet\begin{align*}sqrt{s^2} &= \sqrt{144}\\ s&=12 \ feet\end{align*}

The length of one of the sides of the garden is 12 feet.

Next, we need to figure out the perimeter of the garden.

P=4s\begin{align*}P=4s\end{align*}

Let’s substitute in the known value for the side of the garden.

P=4(12)\begin{align*}P=4(12)\end{align*}

The perimeter of the garden is 48 square feet.

Now let’s go back and figure out the problem from the introduction.

## Real-Life Example Completed

The Food Drive

Here is the problem from the introduction. Reread it and then solve the problem at the end.

The eighth grade student council decided that the theme for the school year would be “Helping Hands.” With this theme, the whole year would focus the eighth graders around different community service projects. When the president of the class, Margaret, proposed this to the student body, the students were all very excited. They decided to let each home room figure out what project they were each going to focus on.

Mrs. Garibaldi’s class held a canned-food drive to aid a local relief shelter. Juan was the team leader. He sent out a notice for each family to begin gathering food in the beginning of November. He figured that they could collect all of their cans by Thanksgiving and provide some families with extra food for the holidays.

They collected 121 cans for the shelter. Many different types of canned food was collected. Juan calculated that the number of cans containing vegetables was equal to 12112+14\begin{align*}121^{\frac{1}{2}}+14\end{align*}.

“How many cans contained vegetables?” Margaret asked Juan at lunch.

Juan simply smiled and wrote his expression on a piece of paper.

“That is a weird way to write it!” she said. “I still don’t know how many cans contained vegetables!”

Now use what you have learned to solve this problem.

Solution to Real – Life Example

To create an equation for this scenario, it is important to first identify the variable. The unknown in this problem is the number of cans that contain vegetables. We can name that quantity c\begin{align*}c\end{align*}.

The problem tells us that c\begin{align*}c\end{align*} is equal to the expression 12112+14\begin{align*}121^{\frac{1}{2}}+14\end{align*}. So we can write an equation with c\begin{align*}c\end{align*} on one side and 12112+14\begin{align*}121^{\frac{1}{2}}+14\end{align*} on the other.

\begin{align*}c=121^{\frac{1}{2}}+14\end{align*}

To solve this equation, you can simplify the exponent and add. We know that when a number is raised to the power of \begin{align*}\frac{1}{2}\end{align*}, it is the same as finding the square root. So, the equation can be rewritten.

\begin{align*}c=\sqrt{121}+14\end{align*}

The square root of 121 is 11, since \begin{align*}11 \times 11=121\end{align*}. So, the equation becomes even simpler.

\begin{align*}c=11+14\end{align*}

Since \begin{align*}11+14=25\end{align*}, the value of \begin{align*}c\end{align*} is 25.

\begin{align*}c=25\end{align*}

The number of cans that contained vegetables in Mrs. Garibaldi’s class was 25.

## Vocabulary

Here are the vocabulary words from this lesson.

Exponent
the little number that represents a power. It tells you how many times to multiply the base by itself.
Base
the number being raised to a power. It is the large number next to an exponent.
a number inside a radical where you will need to find the root of a number.
Squared
an exponent of 2, tells you to multiply the base by itself.
Cubed
an exponent of 3, tells you to multiply the base by itself three times.
Cube Root
to find a value that when multiplied by itself three times is equal to the value inside the radical.
Perfect Square
A number that is a square of a whole number.
Perfect Cube
a number that is the cube of a whole number.
Fractional Power
an exponent in fraction form. A fractional exponent of \begin{align*}\frac{1}{2}\end{align*} is the same as the square root of a number. A fractional exponent of \begin{align*}\frac{1}{3}\end{align*} is the same as the cube root of a number.

## Time to Practice

1. \begin{align*}\overset{}{\overline{ ) {16}}}\end{align*}
2. \begin{align*}\overset{}{\overline{ ) {25}}}\end{align*}
3. \begin{align*}\overset{}{\overline{ ) {81}}}\end{align*}
4. \begin{align*}\overset{}{\overline{ ) {121}}}\end{align*}
5. \begin{align*}\overset{}{\overline{ ) {36}}}\end{align*}
6. \begin{align*}\overset{}{\overline{ ) {169}}}\end{align*}
7. \begin{align*}\sqrt[3]{125}\end{align*}
8. \begin{align*}\sqrt[3]{64}\end{align*}
9. \begin{align*}\sqrt[3]{27}\end{align*}
10. \begin{align*}\overset{}{\overline{ ) {144}}}\end{align*}

Directions: Approximate each square root by listing the two values that the square root can be found between.

1. \begin{align*}\overset{}{\overline{ ) {12}}}\end{align*}
2. \begin{align*}\overset{}{\overline{ ) {15}}}\end{align*}
3. \begin{align*}\overset{}{\overline{ ) {20}}}\end{align*}
4. \begin{align*}\overset{}{\overline{ ) {22}}}\end{align*}
5. \begin{align*}\overset{}{\overline{ ) {31}}}\end{align*}
6. \begin{align*}\overset{}{\overline{ ) {90}}}\end{align*}
7. \begin{align*}\overset{}{\overline{ ) {99}}}\end{align*}

Directions: Evaluate each fractional power.

1. \begin{align*}64^{\frac{1}{3}}\end{align*}
2. \begin{align*}16^{\frac{1}{2}}\end{align*}
3. \begin{align*}144^{\frac{1}{2}}\end{align*}
4. \begin{align*}81^{\frac{1}{2}}\end{align*}
5. \begin{align*}9^{\frac{1}{2}}\end{align*}

Directions: Solve each equation involving radical expressions.

1. \begin{align*}x^2=121\end{align*}
2. \begin{align*}x^3=27\end{align*}
3. \begin{align*}x^2+3=147\end{align*}
4. \begin{align*}x^2-2=23\end{align*}
5. \begin{align*}x^2+11=47\end{align*}

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