Exponent Experimentation 2

Task

Here are some different ways to write the value $16$:

$$2^4$$

$$12-(2^1+2^2)+\frac{500}{50}$$

$$2^3+2^3$$

$$\frac23\times48^1-(1+3)^2$$

Find at least three different ways to write each value below. Include at least one exponent in all of the expressions you write.

1. 81
2. $2^5$
3. $\frac{64}{9}$

 

IM Commentary

The purpose of this task is to give students experience experimenting with equivalent numerical expressions. This work supports fluency because students practice working with operations, decomposing numbers, and recognizing perfect squares and perfect cubes.

Before setting students to work, the sample expressions are an opportunity to discuss order of operations when an expression uses exponents.

In order for students to really flex their thinking about the meaning of exponential expressions, a lesson designed around this task should encourage students to be creative and come up with unique expressions. For example, you might place students in groups of two or three and add the condition that their group many not have any duplicate expressions.

After students generate their expressions, it may be productive to include an error-checking component. For example, after students have had a few minutes to work on part (a), write all unique expressions on the board. Demonstrate the use of a tool like Desmos or a TI-84 with an updated operating system (which format exponential expressions in an obvious way) to check each expression and sort them into "equals 81" and "doesn't equal 81." (Note that this may be counterproductive with tools like a TI-83 or a four-function calculator that don't format expressions the same way you would write them.)

(This example was generated on Desmos.)

Solution

1. Answers will vary. Some expressions that are equivalent to $81$ are $9^2$, $3\times3^3$, and $8^2+17$.
2. Answers will vary. Some expressions that are equivalent to $2^5$ are $32^1$, $\frac{2^6}{2}$, and $4^3\div2$. (Note: the complex fraction $\frac{2^6}{2}$ is notation that students should learn sometime in grade 6. So whether you expect to see this from students may depend on when this task happens in the school year.)
3. Answers will vary. Some expressions that are equivalent to $\frac{64}{9}$ are $8^2\div3^2$, $\frac{1}{3^2}\times2^6$, and $\frac19\times(2^2+60)$.