IESN - Properties of Integer Exponents Lesson

Powers and Exponents

In previous math classes, you have learned about an exponent and a power. Let us take a moment to review before we begin learning about the integer exponents.

Using exponents is a short way to express repeated multiplication. The exponent represents how many times the base is to be used as a factor. The number produced by raising a base to an exponent is called a power. For example, both 16 and 4² represent the same power.

4² 4 represents the base and 2 represents the exponent. 4 raised to the power of 2 is 4 x 4 = 16

Try a few to make sure you remember how to simplify exponential expressions:

5³=

4⁴=

10⁵=

$LaTeX: (\frac{2}{3})^4$ $(\frac{2}{3})^4$ =

5x5x5 = 125

4x4x4x4 = 256

10x10x10x10x10=100,000

$LaTeX: (\frac{2}{3})\times(\frac{2}{3})\times(\frac{2}{3})\times(\frac{2}{3})=(\frac{16}{81})$ $(\frac{2}{3})\times(\frac{2}{3})\times(\frac{2}{3})\times(\frac{2}{3})=(\frac{16}{81})$

Sometimes, it is helpful to expand the exponent to avoid the common mistake of multiplying the base by the exponent.

When the base is negative PARENTHESES are needed. The exponent is “attached” to each parenthesis.

Everything inside the parentheses gets multiplied.

(-2)⁴ = (-2) x (-2) x (-2) x (-2) = 16

(-3)³ = (-3) x (-3) x (-3) = -27

When the expression contains a negative sign with NO parentheses the negative is NOT included as the base. The exponent is ONLY “attached” to the number, NOT the sign.

-2⁴= 2 x 2 x 2 x 2 = 16 x -1 = -16

-3³ = 3 x 3 x 3 = 27 x -1 = -27

Zero Power and Negative Exponents

Now, we will learn to evaluate expressions with negative exponents and with the zero exponent.

Do you remember the term, reciprocal? It is simply 1 divided by that number. In other words, the reciprocal of 7 is 1/7 and the reciprocal of 17 is 1/17. We will use the reciprocal of our exponent when we evaluate expressions with negative exponents. We will also use a phrase “ cross the line and change the sign” to help us remember how to simplify or evaluate numbers with negative exponents.

Words	Numbers	Algebra
Any number except 0 with a negative exponent equals its reciprocal with the opposite exponent.	$LaTeX: 5^{-3}=(\frac{1}{5})^3=\frac{1}{125}$ $5^{-3}=(\frac{1}{5})^3=\frac{1}{125}$	$LaTeX: b^{-n}=(\frac{1}{b})^n,b\ne0$ $b^{-n}=(\frac{1}{b})^n,b\ne0$

When you have a negative exponent, your base crosses the fraction line(cross the line) from numerator to denominator and the sign of the exponent changes to positive. “Cross the Line and Change the Sign”

Examples

1. $LaTeX: 7^{-2}=\frac{1}{7^2}=\frac{1}{7\cdot7}=\frac{1}{49}$ $7^{-2}=\frac{1}{7^2}=\frac{1}{7\cdot7}=\frac{1}{49}$

2. $LaTeX: (-7)^{-2}=\frac{1}{(-7)^2}=\frac{1}{-7\cdot-7}=\frac{1}{49}$ $(-7)^{-2}=\frac{1}{(-7)^2}=\frac{1}{-7\cdot-7}=\frac{1}{49}$

3. $LaTeX: -7^{-2}=\frac{1}{-7^2}=-\frac{1}{49}$ $-7^{-2}=\frac{1}{-7^2}=-\frac{1}{49}$

Try it

Evaluate the following:

2^-6
(-3)^-4
-2^-3
(2/3)^-2

Solutions:

1/64
1/81
-1/8
9/4

Zero Power

Evaluating numbers with zero exponents is the easiest rule in math.

The zero power of any number, except 0, equals 1.

25⁰= 1
(-9)⁰= 1
x⁰= 1 ( as long as x does not equal 0)

Operations with Integer Exponents

Now that we have reviewed the exponent rules, learned to evaluate negative integer exponents and the zero exponent, we are ready to simplify expressions containing integer exponents using the Product Rule, the Quotient Rule, and the Power of a Power Rule. These rules apply only to those powers with the same base.

Let us start with multiplying powers with the same base. The factors of powers can be grouped in different ways, which explains how this property works. The factors of 5⁵ can be grouped as follows:

5 x 5 x 5 x 5 x 5 = 5⁵

(5 x 5 x 5) x 5 x 5 = 5³ x 5²= 5⁵

(5 x 5 x 5 x 5) x 5 = 5⁴ x 5¹= 5⁵

*Notice the relationship of the exponents in each product? It is the same!*

Underneath is the rule and examples for multiplying powers with the same base (Product Rule)

Explanation	Example	Algebra
To multiply powers with the same base, keep the base and add the exponents.	4³+ 4⁷= 4³⁺⁷= 4¹⁰	b^x+ b^y= b^x+y

Now take a look at what is happening when you divide powers with the same base.

$LaTeX: \frac{5^4}{5^2}$ $\frac{5^4}{5^2}$ can be written as $LaTeX: \frac{5\cdot5\cdot5\cdot5}{5\cdot5}$ $\frac{5\cdot5\cdot5\cdot5}{5\cdot5}$ and simplified to 5².

Below is the rule and examples for dividing powers with the same base (Quotient Rule)

Explanation	Example	Algebra
To divide powers with the same base, keep the base and subtract the exponents.	$LaTeX: \frac{8^4}{8^2}=8^2$ $\frac{8^4}{8^2}=8^2$	$LaTeX: \frac{a^x}{a^y}=a^{x-y}$ $\frac{a^x}{a^y}=a^{x-y}$

The third rule we will learn involves raising a power to a power. It looks like this: (4³)². This would be read as four to the third to the second power. If you use the order of operations, you can see how the rule would work. (4 x 4 x 4)²=(4 x 4 x 4) x (4 x 4 x 4) = 4⁶.

Below is the rule and examples for raising a power to a power (Power to a Power Rule)

Explanation	Example	Algebra
To raise a power to a power, keep the base and multiply the exponents.	(8⁴)²= 8^{4 x 2} = 8⁸	(b^m )ⁿ = b^{m x n}

Want a copy of these notes to look over? CLICK HERE Links to an external site. to download a pdf containing the notes and equations

Let's watch this Khan Academy Links to an external site. video to see all of these exponent rules being used:

You're feeling like an expert now, right!!

Now you try some!

Practice

If you would like to practice these types of problems and check your work before you complete your graded homework assignment, click here. Links to an external site. Make sure you check your answers and look at the course resources or ask your teacher if there are any problems you do not understand. Links to an external site.

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