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# Evaluation and Comparison of Powers

## Understand relative effect of powers on value of expressions.

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Evaluation and Comparison of Powers

Did you try to figure out the dimensions of the tiger cage from the last lesson?

Remember, that the tiger cage at the other zoo had dimensions of 183\begin{align*}18^3\end{align*}. This means that the height of the cage was 18 feet, the width of the cage was 18 feet and the length of the cage was 18 feet?

How big was the cage? To figure this out, you will need to evaluate powers. Pay close attention and you will learn how to do this in the following Concept.

### Guidance

Let's review bases and exponents for just a moment.

The large number is called the base. You can think about the base as the number that you are working with.

The small number is called the exponent. The exponent tells us how many times to multiply the base by itself.

An exponent can also be known as a power.

We can read bases and exponents.

35\begin{align*}3^5\end{align*} is read as "three to the fifth power".

Once you know the base and exponent, then you can think about how to evaluate a power. This means that we actually complete the multiplication and figure out the new product.

52

We want to evaluate 5 squared. We know that this means 5×5.\begin{align*}5 \times 5.\end{align*} First, we write it out as factors.

52=5×5

Next, we solve it.

52=5×5=25

RED ALERT!!! The most common mistake students make with exponents is to just multiply the base by the exponent.

52\begin{align*}5^2\end{align*} IS NOT 5×2\begin{align*}5 \times 2\end{align*}

The exponent tells us how many times to multiply the base by itself.

52\begin{align*}5^2\end{align*} is 5×5\begin{align*}5 \times 5\end{align*}

Be sure to keep this in mind!!!

We can also compare the values of powers using greater than, less than and equal to.

We use our symbols to do this.

Greater than >\begin{align*} > \end{align*}

Less than <\begin{align*} < \end{align*}

Equal to =\begin{align*}=\end{align*}

To compare the value of different powers, we will need to evaluate each power and then compare them.

Here are a few examples for you to complete on your own.

#### Example A

26\begin{align*}2^6\end{align*}

Solution: 64

#### Example B

63\begin{align*}6^3\end{align*}

Solution: 216

#### Example C

2753\begin{align*}2^7 \underline{\;\;\;\;\;\;\;}5^3\end{align*}

Solution: >

Now back to the original problem about the tiger cage.

To figure this out, we can take evaluate the power.

183\begin{align*}18^3\end{align*}

18 x 18 x 18 = 5,832 ft^3

This is our solution.

### Vocabulary

Whole number
a number that represents a whole quantity
Base
the whole number part of a power
Power
the value of the exponent
Exponent
the little number that tells how many times we need to multiply the base by itself
Squared
the name used to refer to the exponent 2
Cubed
the name used to refer to the exponent 3

### Guided Practice

Evaluate each of these problems on your own.

1. 1100\begin{align*}1^{100}\end{align*}

2. 19114\begin{align*}1^9 \underline{\;\;\;\;\;\;\;} 1^{14}\end{align*}

3. 4554\begin{align*}4^5 \underline{\;\;\;\;\;\;\;}5^4\end{align*}

1. One to any power is equal to 1.

2. These two values are equal.

3. Four to the fifth power is equal to 1024. Five to the fourth power is equal to 625, Therefore, 4^5 is greater than 5^4.

### Practice

Directions: Evaluate the value of each power.

1. 22\begin{align*}2^2\end{align*}

2. 32\begin{align*}3^2\end{align*}

3. 62\begin{align*}6^2\end{align*}

4. 73\begin{align*}7^3\end{align*}

5. 84\begin{align*}8^4\end{align*}

6. 26\begin{align*}2^6\end{align*}

7. 35\begin{align*}3^5\end{align*}

8. 64\begin{align*}6^4\end{align*}

9. 53\begin{align*}5^3\end{align*}

10. 1100\begin{align*}1^{100}\end{align*}

Directions: Compare each power using <\begin{align*} < \end{align*}, >\begin{align*} > \end{align*}, or =\begin{align*}=\end{align*}

11. 4224\begin{align*}4^2 \underline{\;\;\;\;\;\;\;}2^4\end{align*}

12. 3215\begin{align*}3^2 \underline{\;\;\;\;\;\;\;}1^5\end{align*}

13. 6336\begin{align*}6^3 \underline{\;\;\;\;\;\;\;}3^6\end{align*}

14. 7252\begin{align*}7^2 \underline{\;\;\;\;\;\;\;} 5^2\end{align*}

15. 8392\begin{align*}8^3 \underline{\;\;\;\;\;\;\;} 9^2\end{align*}

### 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.
Power

Power

The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
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.
Whole Numbers

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...