# 9.1: Exponent Rules

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 8, Lesson 2.

In this activity, you will explore:

• expressions involving products and quotients with exponents
• expressions with negative and zero exponents

Before beginning the activity, clear out all functions from the Y=\begin{align*}Y=\end{align*} screen and turn off all stat plots.

## Problem 1 - Discovering Exponent Rules

Run the ExpRules program by pressing PRGM then choosing it from the menu and pressing ENTER.

This program allows you to explore 6 different rules of exponents by helping you evaluate exponential expressions for different values of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}. To begin, choose Experiment, then type 1 to explore Rule 1.

The program displays the expression that you will be calculating to explore Rule 1, 2x\begin{align*}2^x\end{align*}2y\begin{align*}2^y\end{align*}. Calculate the expression several times, choosing values from 1 through 8 for x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}. Make and test a conjecture.

Repeat this process to explore rules 26\begin{align*}2 - 6\end{align*}. Pay attention to the prompts, as some rules require you to enter negative values for the variables. Record your conjectures below.

• Rule 1: Make a rule for the product of two powers with like bases.
• Rule 2: Make a rule for the quotient of two powers with like bases.
• Rule 3: Make a rule for the power of a power.
• Rule 4: Make a rule for a power with a negative exponent.
• Rule 5: Make a rule for a power with a zero exponent.
• Rule 6: Make a rule for the power of a quotient.

## Extension

Use your calculator to evaluate each of the expressions shown.

Then make a conjecture for m1n\begin{align*}m^\frac{1}{n}\end{align*}.

3612=813=4912=1612=1614=\begin{align*}36^\frac{1}{2} = \underline{\;\;\;\;\;\;\;} && 8^\frac{1}{3} = \underline{\;\;\;\;\;\;\;} && 49^\frac{1}{2} = \underline{\;\;\;\;\;\;\;} && 16^\frac{1}{2} = \underline{\;\;\;\;\;\;\;} && 16^\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

Complete: m1n=\begin{align*}m^\frac{1}{n} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

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