Have you ever collected can goods for charity? Take a look at this dilemma.
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 cannedfood 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 \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 Concept will work with fractional exponents and radicals. At the end of it, you will know how to help Margaret figure things out.
Guidance
You can evaluate a radical expressions by evaluating the exponent or the root.
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.
\begin{align*}5^2\end{align*}
This means that we multiply the base of 5 by itself two times. This is how we evaluate a power.
\begin{align*}5 \times 5 = 25\end{align*}
This is the answer.
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, \begin{align*}\sqrt{49}\end{align*}
Now let’s look at how we can work with roots and radicals.
\begin{align*}5^2 = 5 \times 5 = 25\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.
\begin{align*}\sqrt{25}\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 5 because 5 squared is equal to 25.
Consider this statement of equivalence.
\begin{align*}7^2 &= 49\\ \sqrt{49}&=7\end{align*}
There is also equivalence when we use radical expressions and fractional powers. Take a look.
There is 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 \begin{align*}\frac{1}{3}\end{align*}
Take a look at this situation.
Elena was asked to find the value of \begin{align*}27^{\frac{1}{3}}\end{align*}
The first step Elena should take is to convert the fractional exponent to a root. Since the fraction is \begin{align*}\frac{1}{3}\end{align*}
\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 \begin{align*}3 \times 3 \times 3\end{align*}
\begin{align*}27^{\frac{1}{3}}=\sqrt[3]{27}=3\end{align*}
The answer is 3.
Evaluate each example.
Example A
\begin{align*}64^{\frac{1}{3}}\end{align*}
Solution: \begin{align*}4\end{align*}
Example B
\begin{align*}49^{\frac{1}{2}}\end{align*}
Solution: \begin{align*}7\end{align*}
Example C
\begin{align*}343^{\frac{1}{3}}\end{align*}
Solution: \begin{align*}7\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
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 \begin{align*}c\end{align*}
The problem tells us that \begin{align*}c\end{align*}
\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*}
\begin{align*}c=\sqrt{121}+14\end{align*}
The square root of 121 is 11, since \begin{align*}11 \times 11=121\end{align*}
\begin{align*}c=11+14\end{align*}
Since \begin{align*}11+14=25\end{align*}
\begin{align*}c=25\end{align*}
The number of cans that contained vegetables in Mrs. Garibaldi’s class was 25.
Vocabulary
 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.
 Radical Expression
 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*}
12 is the same as the square root of a number. A fractional exponent of \begin{align*}\frac{1}{3}\end{align*}13 is the same as the cube root of a number.
Guided Practice
Here is one for you to try on your own.
Birgit needs to solve the equation \begin{align*}x=81^{\frac{1}{2}}\end{align*}
Solution
Birgit should realize that raising something to the power of \begin{align*}\frac{1}{2}\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 \begin{align*}9 \times 9=81\end{align*}
In the equation \begin{align*}x=81^{\frac{1}{2}}, x=9\end{align*}
Video Review
Khan Academy Fractional Powers
Practice
Directions: Evaluate each fractional power.

\begin{align*}64^{\frac{1}{2}}\end{align*}
6412  \begin{align*}16^{\frac{1}{2}}\end{align*}
 \begin{align*}144^{\frac{1}{2}}\end{align*}
 \begin{align*}81^{\frac{1}{2}}\end{align*}
 \begin{align*}9^{\frac{1}{2}}\end{align*}
 \begin{align*}25^{\frac{1}{2}}\end{align*}
 \begin{align*}216^{\frac{1}{3}}\end{align*}
 \begin{align*}100^{\frac{1}{2}}\end{align*}
 \begin{align*}16^{\frac{1}{4}}\end{align*}
 \begin{align*}256^{\frac{1}{4}}\end{align*}
 \begin{align*}125^{\frac{1}{3}}\end{align*}
 \begin{align*}36^{\frac{1}{2}}\end{align*}
 \begin{align*}81^{\frac{1}{4}}\end{align*}
 \begin{align*}121^{\frac{1}{2}}\end{align*}
 \begin{align*}169^{\frac{1}{2}}\end{align*}