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# Fractional Exponents

## Relate fractional exponents to nth roots

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Practice Fractional Exponents
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Evaluate Radical Expressions and Fractional Powers

Have you ever collected canned goods for charity?

The eighth grade student council decided that the theme for the school year would be “Helping Hands.” With this theme, the eighth graders would focus on different community service projects throughout the year. 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 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 were collected. Juan counted the number of cans containing vegetables.

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

Juan simply smiled and wrote this expression on a piece of paper: ${121}^{\frac{1}{2}}+14$ .

“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.

$5^2$

This means that we multiply the base of 5 by itself two times. This is how we evaluate a power.

$5 \times 5 = 25$

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, $\sqrt{49}$ , 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.

$5^2 = 5 \times 5 = 25$

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.

$\sqrt{25}$

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.

$7^2 &= 49\\\sqrt{49}&=7$

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 $\frac{1}{3}$ , it is the same as taking the cube root. Similarly, if a number is raised to the power of $\frac{1}{2}$ , it is the same as taking the square root.

Take a look at this situation.

Elena was asked to find the value of $27^{\frac{1}{3}}$ . 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 $\frac{1}{3}$ , she will need to find the cube root of 27 to solve the problem.

$27^{\frac{1}{3}}=\sqrt[3]{27}$

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

If you multiply $3 \times 3 \times 3$ , the product is 27. So, the cube root of 27 is 3.

$27^{\frac{1}{3}}=\sqrt[3]{27}=3$

The answer is 3.

Evaluate each example.

#### Example A

$64^{\frac{1}{3}}$

Solution:  $4$

#### Example B

$49^{\frac{1}{2}}$

Solution:  $7$

#### Example C

$343^{\frac{1}{3}}$

Solution:  $7$

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 $c$ .

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

$c=121^{\frac{1}{2}}+14$

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

$c=\sqrt{121}+14$

The square root of 121 is 11, since $11 \times 11=121$ . So, the equation becomes even simpler.

$c=11+14$

Since $11+14=25$ , the value of $c$ is 25.

$c=25$

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

### Guided Practice

Here is one for you to try on your own.

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

Solution

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

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 $9 \times 9=81$ , the square root of 81 is 9.

In the equation $x=81^{\frac{1}{2}}, x=9$ .

### Explore More

Directions: Evaluate each fractional power.

1. $64^{\frac{1}{2}}$
2. $16^{\frac{1}{2}}$
3. $144^{\frac{1}{2}}$
4. $81^{\frac{1}{2}}$
5. $9^{\frac{1}{2}}$
6. $25^{\frac{1}{2}}$
7. $216^{\frac{1}{3}}$
8. $100^{\frac{1}{2}}$
9. $16^{\frac{1}{4}}$
10. $256^{\frac{1}{4}}$
11. $125^{\frac{1}{3}}$
12. $36^{\frac{1}{2}}$
13. $81^{\frac{1}{4}}$
14. $121^{\frac{1}{2}}$
15. $169^{\frac{1}{2}}$

### Vocabulary Language: English

Base

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$, 32 is the base, and 4 is the exponent.
Cubed

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = $2^3 = 2 \times 2 \times 2 = 8$.
Exponent

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Fractional Power

Fractional Power

A fractional power is an exponent in fraction form. A fractional exponent of $\frac{1}{2}$ is the same as the square root of a number. A fractional exponent of $\frac{1}{3}$ is the same as the cube root of a number.
Perfect Square

Perfect Square

A perfect square is a number whose square root is an integer.
Radical Expression

Radical Expression

A radical expression is an expression with numbers, operations and radicals in it.
Squared

Squared

Squared is the word used to refer to the exponent 2. For example, $5^2$ could be read as "5 squared". When a number is squared, the number is multiplied by itself.

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