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Grade 9:  The Learning Equation Math

11.04:  Powers, Bases and Exponents

Numbers

Number Concepts

Refresher pp 8-9
 

 

SeekAWord2ech.class Author

Prerequisite Skills:

 

Key Terms                 

power, exponent, base, coefficient

Interactive Component Under Construction

 

Learning Outcomes

The student will:

 

Powers

Click on to get the value of 32.

Enter the exponent and base, then click on to calculate other powers.    The exponent on this calculator must be less than 600.
Enter the Exponent:

Enter the Base:

( )

=

Note:  00 = indeterminate.  The answer shown for 00 on this calculator is incorrect.

Source:  Web Winder JavaScript Calculators

 

Exponential Form Other Expanded Form(s)/Repeated Multiplication(s) Other Exponential Form(s)
45 = (22)5 22 x 22 x 22 x 22 x 22 22+2+2+2+2 = 210

84 = (23)4

23 x 23 x 23 x 23

23+3+3+3 = 212
93 = (32)3 32 x 32 x 32 32+2+2 = 36
163 = (42)3 = (24)3
 42 x 42 x 42 (42)3 = 42x3 = 46
(24) x (24) x (24) (24)3 = 44x3 = 412

252 = (52)2

 52 x 52 52+2 = 54

273 = (33)3

 33 x 33 x 33 33+3+3 = 39

 

 

Rational Numbers and Variables

Power Coefficient Base Exponent

3.3x4

 3.3 x 4

2/3a3

 2/3 a 3
-2.5n2 -2.5 n 2
4.5z or  4.5z1 4.5 z 1
x -2 1 x -2

p r 2 is an example where an irrational numbers (p) is used as a coefficient.

Powers can be expressed using rational numbers and variables as bases and/or coefficients.  If there is no operation sign (+, -, x, /), it is understood the operation is multiplication.

4 x t x t x t x t = 4tttt

Exponential Form (Power) Expanded Form/Repeated Multiplication Simplified Expression

3n5

3 x n x n x n x n x n

or

3nnnnn

Cannot be simplified.

2.5r3

 2.5rrr

Cannot be simplified
(2/3r)4 (2/3r)(2/3r)(2/3r)(2/3r) 16/81r4
(-2/3r)4 (-2/3r)(-2/3r)(-2/3r)(-2/3r) 16/81r4
-(2/3r)4 -(2/3r)(2/3r)(2/3r)(2/3r) -16/81r4

 

Scientific Calculator

Press Reload/Refresh if calculator does not function.
Follow the directions to learn how to convert from scientific notation to whole numbers using the scientific calculator, then make your own examples.
-54 = - ( 5 wpeB1.jpg (772 bytes) wpeB6.jpg (768 bytes) ) = -625
(-5)4 = 5 wpe5A.jpg (777 bytes) wpeB1.jpg (772 bytes) wpeB6.jpg (768 bytes)  = 625
- (-5)4 =  5 wpe5A.jpg (777 bytes) wpeB1.jpg (772 bytes) wpeB6.jpg (768 bytes) wpe5A.jpg (777 bytes) = -625
(-5)0 5 wpe5A.jpg (777 bytes) wpeB1.jpg (772 bytes) 0.gif (941 bytes) = 1
00 Note:   00 = indeterminate.  The answer shown for 00 on this calculator is incorrect.
Exponential Form (Power) Expanded Form/Repeated Multiplication

-3m6

-3mmmmmm

2.5x2

 2.5xx
(1/3k)5 (1/3k)(1/3k)(1/3k)(1/3k)(1/3k)
(-1/3p)5 (-1/3p)(-1/3p)(-1/3p)(-1/3p)(-1/3p)
-(1/3b)5 -(1/3b)(1/3b)(1/3b)(1/3b)(1/3b)

Question Generator

  • ^ refers to an exponent.  3^3 refers to 32.
  • use / to represent divide and * to represent multiply.
  • use + and - for add and subtract.
  • any letter can be used as a variable.
    •

Click to enter an expression, then click the button.   Try to calculate the answer mentally while the answer page loads.   The Power Property rules in the chart above may be useful.

 

Doubling Bacteria

Number of Ancestors 1000 Years Ago

Problem Solving Steps

1.  Think about the problem
2.  Make a plan including the problem solving strategy
3.  Solve the problem.
4.  Look back.
  • Reread the question.
  • Did you answer the question asked?
  • Is your answer in the correct units?
  • Does your answer seem reasonable?
5.  Look ahead.

 

Problem Solving Strategies

sequence the operations
use a flow chart
use a table
look for a pattern
guess and check
solve a simpler problem (break into parts)
work backward
interpret graphs
use a diagram/sketch 
use a formula
make assumptions
use a data bank
use logic
 

Optional resources:

Curriculum Support for Math- Problem Solving Part 2

Problem Solving Strategies

problem solving menu

Sequencing the Operation

1.  List the facts.
2.  Decide on the solution order.
3.  Make the calculation.
4.  Check the answer

problem solving menu

Use a Flow Chart

1.  Use a flow chart to represent the problem.
2.  Solve the problem.
3.  Check the answer.

problem solving menu

Use a Table

1.  Organize the data in the table.
2.  Complete the table if necessary.
3.  Find the answer from the table.
4.  Check the answer.

problem solving menu

Look for a Pattern

1.  Study the information you are given on the graph.
2.  Determine what you are to find.
3.  Determine if you need an approximate or exact answer.

problem solving menu

Guess and Check

1.  Study the problem and guess the answer.
2.  Check your guess.
3.  If necessary guess and check again. 

problem solving menu

Solve a Simpler Problem (Break into Parts)

1.  Break the problem into smaller problems.
2.  Solve the problem.
3.  Check the answer.

problem solving menu

Work Backward

1.  Start with what you already know.
2.  Work backward to get your answer.
3.  Check the answer.

problem solving menu

Interpret Graphs

1.  Study the information you are given on the graph.
2.  Determine what you are to find.
3.  Determine if you need an approximate or exact answer.

problem solving menu

Use a Diagram/Sketch

1.  Study the information you are given on the graph.
2.  Determine what you are to find.
3.  Determine if you need an approximate or exact answer.

problem solving menu

Use a Formula

1.  Write the formula.
2.  Substitute variables with know values.
3.  Calculate for the remaining variable.
4.  Check the answer

problem solving menu

Make Assumptions

1.  Study the information you are given on the graph.
2.  Determine what you are to find.
3.  Determine your assumptions.

problem solving menu

Use a Data Bank

1.  Look up the information you need.
2.  Solve the problem.
3.  Check the answer.

problem solving menu

Use Logic/Reasoning

1.  Organize the data.
2.  Interpret the data to find the answer.
3.  Check the answer.

problem solving menu

 

Review:

Powers Review

Powers are calculated by multiplying the base by itself the number of times indicated by the exponent.  As multiplication is to repeated addition, powers are to repeated multiplication.

power = baseexponent = base^exponent

Special names are used for exponent two and three.

base2 =base x base = base^2 = base squared

base3 = base x base x base = base^3 = base cubed

Concept/Term Example(s) Definition/Explanation
power

 81 is the fourth power of 3, because:

3 x 3 x 3 x 3 = 81

34 is another way of writing this power

 the answer from multiplying a number by itself one or more times.

A power has two parts:

baseexponent = 34 

and can be written/read in several ways:

  • the fourth power of three
  • three exponent four
  • three to the fourth power

32

33

 Exponents 2 and 3 can also be written/read another way:
  • 32  is read three squared
  • 33  is read three cubed
base

In 34 , the base is 3.

 the number being multiplied in a power
exponent In 34 , the exponent is 4. the number of times a number is being multiplied by itself in a power
coefficient In 5x2, the coefficient is 5. the constant (number) multiplier of the variables in a term

 

Exponential Form (Power) Expanded Form/Repeated Multiplication Standard Form (Standard Name)

34

3 x 3 x 3 x 3

81

33

 3 x 3 x 3 27
32 3 x 3 9
31 3 3
30 no expanded form 1

Note:  The standard name of any non-zero base with an exponent of zero is equal to 1.

Positive Integer Exponents

Exponents - contains review and enrichment

 

Enrichment:

Powers of x

Exponents - contains review and enrichment

 
 

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Comments to:  Jim Reed
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