Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

1. Complete the table by entering the values for a, b, and c below the table. Take advantage of any patterns you notice.

$x$	4	3	2	1	0
$3^{x}$	81	27	a. _____	b. _____	c. _____

a. ________

Compare your answer:

9

b. ________

Compare your answer:

3

c. ________

Compare your answer:

1

2. Here are some equations. Find the solution to each equation using what you know about exponent rules. Be prepared to explain your reasoning.

a. $9^{x} \cdot 9^{7} = 9^{7}$

Compare your answer:

0

b. $\frac{9^{12}}{9^{x}} = 9^{12}$

Compare your answer:

0

3. Answer the following questions.

a. What is the value of $5^{0}$ ?

Compare your answer:

For positive whole numbers, $5^{a} \cdot 5^{b} = 5^{a + b}$ . For this law to still hold with 0 as an exponent, $5^{0}$ needs to be equal to 1 because $5^{0 + 4} = 5^{4}$ and $1 \cdot 5^{4} = 5^{4}$ .

b. What about $2^{0}$ ?

Compare your answer:

For positive whole numbers, $2^{a} \cdot 2^{b} = 2^{a + b}$ . For this law to still hold with 0 as an exponent, $2^{0}$ needs to be equal to 1 because $2^{0 + 4} = 2^{4}$ and $1 \cdot 2^{4} = 2^{4}$ .

Are you ready for more?

Extending Your Thinking

We know, for example, that $(2 + 3) + 5 = 2 + (3 + 5)$ and $2 \cdot (3 \cdot 5) = (2 \cdot 3) \cdot 5$ . The grouping with parentheses does not affect the value of the expression.

Is this true for exponents? That is, are the numbers $2^{(3^{5})}$ and ${(2^{3})}^{5}$ equal? If not, which is bigger?

Compare your answer:

The two numbers are not even close! Since $3^{5} = 243$ , the number $2^{(3^{5})}$ is $2^{243}$ . On the other hand, ${(2^{3})}^{5} = 2^{3} \cdot 2^{3} \cdot 2^{3} \cdot 2^{3} \cdot 2^{3} = 2^{15}$ . The number $2^{243}$ is much larger than $2^{15}$ .

Which of the two would you choose as the meaning of the expression $2^{3^{5}}$ written without parentheses?

Compare your answer:

Typically, we define repeated exponents by working down the tower, e.g., $2^{3^{5}} = 2^{(3^{5})}$ . One reason for this is that evaluating up the tower, as in ${(2^{3})}^{5}$ , can already be simplified by using algebra rules (in this case, to get $2^{15}$ ).

Self Check

Santos and Sara are checking their work on the following problem: $4\left(2\cdot3\cdot5\right)^0$ .

Santos says the answer is 1. Sara says the answer is 4. Who is correct, and why?

Sara is correct because you first would simplify the parenthesis, and since it's raised to the zero power, it would be 1. Then you would multiply that by 4, for a final answer of 4.
Both Santos and Sara are correct since there are two methods to solve the problem.
Neither Santos nor Sara did the problem correctly.
Santos is correct because anything raised to the zero power is 1.

Additional Resources

Zero as an Exponent

Why is $2^{0} = 1$ and $5^{0} = 1$ , but $2^{3}$ is not the same as $5^{3}$ ?

Answering this question will help us understand the zero exponent rule.

Exponents in general represent repeated multiplication, much like multiplication represents repeated addition. Let’s use that understanding to evaluate $2^{3}$ and $5^{3}$ :

$2^{3} = 2 \cdot 2 \cdot 2 = 4 \cdot 2 = 8$

$5^{3} = 5 \cdot 5 \cdot 5 = 25 \cdot 5 = 125$

8 and 125 are obviously very different, but the method to get to each answer was the same. If we rewrite it as a table, we can work backward to understand how $2^{0}$ and $5^{0}$ are both 1.

Using $y = 2^{x}$ and filling in the parts we have found results in Table 1:

$x$	0	1	2	3
$y$		2	4	8

Repeating that process for $y = 5^{x}$ results in Table 2:

$x$	0	1	2	3
$y$		5	25	125

Since exponents represent repeated multiplication in both tables, additional terms are found by multiplying again by the base. What if we worked backward? If you multiply to move to the right in the table, you can divide by the common ratio to work backward. Start with the 8, and work backward to fill in the table:

$x$	0	1	2	3
$y$	$2 \div 2 = 1$	$4 \div 2 = 2$	$8 \div 2 = 4$	8

Repeat the process for Table 2:

$x$	0	1	2	3
$y$	$5 \div 5 = 1$	$25 \div 5 = 5$	$125 \div 5 = 25$	125

After you have worked backward in each table, you will see that $2^{0}$ and $5^{0}$ both equal 1. When 1 is the exponent, your answer is the same as the common ratio, so when you divide by the common ratio, you end up with 1 as your answer every time. This is an important rule that can save a lot of time.

Try it

Try It: Zero as an Exponent

Find the value of each of the following:

$3^{0}$
$1, 246^{0}$
${(2 \cdot 3 \cdot 7)}^{0}$

Compare your answer:

Here is how to use the zero exponent rule to determine the value of each:

The answer to each expression is 1. If you want to work backward, it doesn’t matter how large the original number is. $3 \div 3 = 1$ and $1246 \div 1246 = 1$ , and both expressions only contain a base and zero as exponent, so the answer is 1. For c, there is an expression inside the parentheses that you can evaluate first, but since it is still the base raised to a zero exponent, the answer will still be one.

5.4.2 Zero Exponent Rule

Activity

Are you ready for more?

Extending Your Thinking

Additional Resources

Zero as an Exponent

Try It: Zero as an Exponent