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OPERATIONS WITH NUMBERS AND EXPONENTS

Unit Overview
This unit begins with a review of real numbers, their properties, and the order of operations.  In addition, the various properties of integer exponents are reviewed and extended to include rational exponents. Using the property of exponents and rational exponents, expressions in radical form will be rewritten in exponential form and vice versa.


Operations with Numbers

Types of Numbers

natural numbers:                   1, 2, 3, ….

whole numbers:                     0, 1, 2, 3, …

integers:                                …–2, –1, 0, 1, 2, …

rational numbers:                   where p and q are integers and 0

Quicktime_Video_Icon  Rational Numbers -- Recipes (02:54) 

irrational numbers:      numbers whose decimal part does not terminate or repeat

real numbers:                all rational and all irrational numbers

Quicktime_Video_Icon  Irrational Numbers -- Travel (02:22) 



Properties of Real Numbers


Order of Operations

To simplify algebraic expressions you must use an order of operations.

Parentheses, Exponents, Multiply, Divide, Add, Subtract

“Please Excuse My Dear Aunt Sally”

          *If multiplication and division are the only two operations, work the problem from left to right

          *If addition and subtraction are the only two operations, work the problem from left to right.         

Quicktime_Video_Icon  Introduction (02:08) 

Quicktime_Video_Icon  Simple Orders--Roller Coaster Capacity (02:33) 

Quicktime_Video_Icon  More Orders--Revenue (03:06)

Quicktime_Video_Icon  Exponents--Around the Loop (03:02) 

Stop!  Go to Questions #1-4 about this section, then return to continue on to the next section.


Properties of Exponents




Quicktime_Video_Icon  Multiplying with Like Bases (01:19) 


Quicktime_Video_Icon  Dividing with Like Bases (01:28) 

Quicktime_Video_Icon  Multiplying Expressions with Like Bases (01:53)

Quicktime_Video_Icon  Dividing Expressions with Like Bases (01:56)

Quicktime_Video_Icon  Raising a Power to a Power (02:01) 

Quicktime_Video_Icon  Raising a Power to a Power in Rational Expressions (02:54) 

Stop!  Go to Questions #5-12 about this section, then return to continue on to the next section.


Rational Exponents

Rational exponents are exponents that are fractions. 

Rational exponents are an alternate way to express roots and can be very useful when dealing with more complicated expressions.

First, let's review the terms associated with radicals.

Now, let's take a look at what a rational exponent is.


Practice:  Write in radical form, and then answer the following questions.


question1What is the index of the radical?

The index is 7.

"Click here" to check the answer.



question1What is the power of x under the radical?

The power of x is 3.

"Click here" to check the answer.

                Solution: 


Practice:  Write in exponential form, and then answer the following questions.

question1What is base of the expression?

The base is 5y.

"Click here" to check the answer.



question1What is rational exponent of the expression?

The exponent is 1 ⁄ 2.

"Click here" to check the answer.


                  Solution:   (The index of the radical is understood to be 2.)


Practice:  Simplify and then answer the following questions.

question1What do you do with the exponents?

Add the exponents.

"Click here" to check the answer.



question1What is the simplified expression?

Solution: x1 = x

"Click here" to check the answer.




Practice:  Simplify and then answer the following questions.

question1What do you do with the exponents?

Subtract the exponents.

"Click here" to check the answer.



question1What is the simplified expression?

Solution: n(4 ⁄ 8) = n(1 ⁄ 2)

"Click here" to check the answer.


Practice:  Simplify and then answer the following questions.

question1What do you do with the exponents?

Multiply the exponents.

"Click here" to check the answer.



question1What is the simplified expression?

Solution: b(12 ⁄ 4) = b3

"Click here" to check the answer.




Quicktime_Video_Icon  Negative Exponents Problems


Practice:  Simplify and then answer the following questions.

question1What is the coefficient of the expression in the numerator?

8 to the (1 ⁄ 3) means cube root of 8 which equals 2.

"Click here" to check the answer.



question1What is the entire expression in the numerator of the solution?

2k(3 ⁄ 3) = 2k1 = 2k

"Click here" to check the answer.



question1What is the entire expression in the denominator of the solution?

m(6 ⁄ 3) = m2


"Click here" to check the answer.


*Remember:  The negative exponent in the numerator becomes positive when put in the denominator of the fraction.   



Practice:  Simplify and then answer the following questions.

question1What is the coefficient of the solution?

23 = 8

"Click here" to check the answer.



question1What is the exponent of y in the solution?

The exponent of y is (2 ⁄ 3)(3 ⁄ 1) = 6 ⁄ 3 = 2

"Click here" to check the answer.




question1What is the solution to the expression?

8y2

"Click here" to check the answer.


Quicktime_Video_Icon  Fractional Exponents (06:21) 


Practice:  Simplify and then answer the following questions.  Use the shortcut to simplify your work.

question1What is the numerator of the solution?

4y3 (The y is moved to the numerator with a positive power.)

"Click here" to check the answer.



question1What is the denominator of the solution?

3x2 (The x is moved to the denominator with a positive power.)

"Click here" to check the answer.



Stop!  Go to Questions #13-29 to complete this unit.





Below are additional educational resources and activities for this unit.
 
Click on the icon to find and practice topics for this unit.
 
Advanced Order of Operations
 
Simplifying Rational Expressions Worksheet 1
 
Simplifying Rational Expressions Worksheet 2