Getting Started

The student does not understand that a nonzero number raised to the zero power is equal to one.

The student states that:

• is 0 and is 0.
•  is 3 and  is 4.
  
  

What does  mean? Can you write this in another way?

Can you explain how = 0 (or  = 3) would fit into the pattern shown on the table?

Review the meaning of positive integer exponents. Using bases with whole number exponents less than ten, model rewriting the expression in an expanded form (e.g., rewrite as 3 x 3 x 3 x 3). Provide the student with sample problems of this type and ask the student to rewrite in expanded form and evaluate. Then provide examples in expanded form and ask the student to rewrite in exponential form.

Consider implementing the MFAS tasks associated with 5.NBT.1.2 and 6.EE.1.1.

Present the student with a table similar to the one used in this task that contains powers of two from to . Assist the student in identifying the patterns shown in the table as the powers decrease (e.g., for any two consecutive entries, as the exponent decreases by one, the value of the exponential expression decreases by a factor of two). Make explicit that any given entry (e.g., 16) can be divided by two (16 ÷ 2) to get the entry (8) for the next lower power of two . Ask the student to divide the entry for  by two to determine the entry for . Finally, ask the student to: 1) construct a table with a different base and 2) construct an explanation for why raising a (nonzero) number to the zero power should be one.

Moving Forward

The student is unable to write equivalent numerical expressions for numbers with negative exponents.

The student understands that and  are both equal to one. However, the student evaluates  and  incorrectly. The student evaluates:

• as 3 and as 9.
  
  
  
  
• as and as .
  
  
  
  
• as -3 and as -9.
  
  
  
  
• as -3 and as -6.
  
  

What pattern do you see in the second column of the table? How is 27 related to 9? How is 9 related to 3? Can you extend that pattern to determine what  should equal?

Present the student with a table similar to the one used in this task that contains powers of two from to . Ask the student to identify the patterns shown in the table as the powers decrease (e.g., for two consecutive entries, as the exponent decreases by one, the value of the exponential expression decreases by a factor of two). Make explicit that any given entry (e.g., 16) can be divided by two (16 ÷ 2) to get the entry (8) for the next lower power of two . Ask the student to extend the pattern in the table to by dividing the previous entry by two. Have the student continue in this manner to determine the values of and . Then ask the student to correct any errors made on the Exponents Tabled worksheet and create his or her own table using a different base.

Model the process of writing numbers with negative integer exponents in an equivalent form using only positive exponents, expanding, and then evaluating (e.g., ). Ask the student to evaluate numbers with negative integer exponents in a similar manner.

Almost There

The student’s explanation for why a number raised to the zero power should be one is incomplete.

The student correctly evaluates  as one, as , as , and  as one. However, when explaining why , the student:

• States that anything to the power of zero is one or refers to the Zero Power Property.
  
  
  
  
• States that four has nothing to multiply itself by.

It is true that any number raised to the zero power is one, but why is it one rather than zero?

Can you identify any patterns in the table to suggest what a number with an exponent of zero should be?

Assist the student in observing patterns in the table to construct an explanation. In addition, provide the student with explanations based on the need to be consistent with the application of previously learned properties of exponents. For example, remind the student of the property (where a is a nonzero rational number and n and m are positive integers). Applying this property to x , yields . So, must be equal to one.

Got It

The student provides complete and correct responses to all components of the task.

The student correctly evaluates  as one,  as ,  as , and  as one. To explain why should equal one, the student refers to:

• The pattern in the values of the powers of three in the table and applies it to a base of four. 
• The properties of multiplying and dividing integer exponents. For example, the student shows that  and concludes that must be equal to one.

Can you evaluate ?

Can you evaluate ?

Provide the student with more challenging problems involving negative integer exponents that include negative bases and fractional bases. Challenge the student to rewrite in an equivalent form using positive exponents.