Su Chin keeps her turtle, Larry, in a glass tank that is 14^{3} cubic inches. Larry loves to eat and is growing at an alarming rate, and Su Chin will soon need a larger tank for him. She looks at an online pet store and sees that they have two tanks for sale: one is 2,744 cubic inches, and the other is 4,096 cubic inches. How can Su Chin compare the size of her tank to the new tanks to ensure she buys a larger one?

In this concept, you will learn how to evaluate and compare powers.

### Powers

\begin{align*}\huge{4^2}\end{align*}

Let's consider this expression.

The large number, 4, is the **base**. The base is the number to be multiplied by itself.

The small number, 2, is the **exponent**. The exponent tells you how many times to multiply the base by itself. The exponent is also known as a **power.**

Once you know the base and exponent, you can evaluate the expression. Evaluating an expression with an exponent means to complete the indicated multiplication and write the result as a product.

Consider the expression above,

.First, identify the base, in this case 4.

Then, identify the exponent, in this case 2.

Next, write the expression as the base multiplied by itself the number of times indicated by the exponent:

\begin{align*}4^2 = 4 \times 4\end{align*}

Finally, perform the multiplication.

\begin{align*}4^2 = 4 \times 4 = 16\end{align*}

**NOTE**: A common error related to exponents is to multiply the base by the exponent. Don't do that! Clearly, since \begin{align*}4^2 = 4 \times 4 \ = 16\end{align*}

You can also compare the values of powers using symbols: \begin{align*}>\end{align*} (greater than), \begin{align*}<\end{align*} (less than) and \begin{align*}=\end{align*} (equal to).

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

Let's look at an example.

Evaluate

This looks a bit intimidating, since the power is so large. However, since the base is 1, and regardless how many times you multiply 1 by itself, it is always 1, this is actually not difficult at all.

The answer is:

In fact, 1 to any power is equal to 1.

### Examples

#### Example 1

Earlier, you were given a problem about Su Chin and her rapidly growing reptile.

Su Chin needs to compare her current tank, which is 14^{3} cubic inches, with two possible replacements that measure 2,744 cubic inches, and 4,096 cubic inches, in order to choose the largest tank.

First, evaluate the power of Su Chin’s current tank.

14^{3} = 14 x 14 x 14 = 2,744

The answer is 2,744 cubic inches.

Then, compare those dimensions to the other tanks, using < , > , or =.

2,744 = 2,744

2,744 < 4,096

The answer is the second tank is largest.

The first tank is the same size as Su Chin’s current tank, at 2,744 cubic inches. The second tank is greater in size, at 4,096 cubic inches.

Su Chin can now buy the second tank in order to upgrade Larry’s home to a spacious 4,096 cubic inches.

#### Example 2

Insert > , < or = in the blank.

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

First, evaluate each number.

\begin{align*}4^5 = & 4 \times 4 \times 4 \times 4 \times 4\\ = & 16 \times 4 \times 4 \times 4\\ = & 64 \times 4 \times 4\\ = & 256 \times 4\\ = & 1024\\ \\ 5^4= & 5 \times 5 \times 5 \times 5\\ = & 25 \times 5 \times 5\\ = & 125 \times 5\\ = & 625\\\end{align*}Then, compare the values. 1024 > 625

The answer is \begin{align*}4^5 > 5^4\end{align*}

#### Example 3

Evaluate the expression

.First, identify the base, in this case 2

Then, identify the exponent, in this case 6.

Next, consider the expression as the base multiplied by itself the number of times indicated by the exponent:

\begin{align*}2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\end{align*}Finally, perform the multiplication.\begin{align*}2^6 = & 2 \times 2 \times 2 \times 2 \times 2 \times 2\\ = & 4 \times 2 \times 2 \times 2 \times 2\\ = & 8 \times 2 \times 2 \times 2\\ = & 16 \times 2 \times 2\\ = & 32 \times 2\\ = & 64\\\end{align*} The answer is 64.

#### Example 4

Evaluate the expression

.First, identify the base, in this case 6.

Then, identify the exponent, in this case 3.

Next, consider the expression as the base multiplied by itself the number of times indicated by the exponent:

\begin{align*}6^3 = 6 \times 6 \times 6\end{align*}Finally, perform the multiplication.\begin{align*}6^3 = & 6 \times 6 \times 6\\ = & 36 \times 6\\ = & 216\\\end{align*}The answer is 216.

#### Example 5

Which is greater?

First, evaluate each number.\begin{align*}2^7 = & 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\\ = & 4 \times 2 \times 2 \times 2 \times 2 \times 2\\ = & 8 \times 2 \times 2 \times 2 \times 2\\ = & 16 \times 2 \times 2 \times 2\\ = & 32 \times 2 \times 2\\ = & 64 \times 2\\ = & 128\\ \\ 5^3= & 5 \times 5 \times 5\\ = & 25 \times 5\\ = & 125\\\end{align*}

Finally, compare the values. 128 > 125.

The answer is \begin{align*}2^7 > 5^3\end{align*}.

### Review

Evaluate each expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.9.