Directions: Part I—Read the following informational text ...



Directions: Part I—Read the following informational text to help you answer the questions that follow regarding multiplying and dividing numbers that have exponents.

Informational Text: Exponents have a few rules or properties that can be used for simplifying expressions. What follows are general rules applied using the variable, x, to signify “any number.” After reading, answer the questions that follow.

Case 1: Multiplying numbers with exponents that have the same base

Example: Simplify (x3)(x4)  

To simplify this product, think in terms of what the exponents mean: "x to the third" means "multiplying the base, x, three times" and "x to the fourth" means "multiplying the base, x, four times." Using this fact, "expand" the two factors, and then work backwards to the simplified form:

(x3)(x4) = (xxx)(xxxx) = xxxxxxx = x7

Note that x7 also equals x(3+4). This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents. Try several other examples to convince yourself. This can be written in general terms as follows:

( x m ) ( x n ) = x( m + n )

Note: You can NOT simplify (x4)(y3), because the bases are different: (x4)(y3) = xxxxyyy = (x4)(y3). Nothing combines. You can only use these properties if the bases are the same.

Case 2: Raising numbers with exponents to a power

Example: Simplify (x2)4

Just as with the previous exercise, think in terms of what the exponents mean. The "to the fourth" means that I'm multiplying four copies of what is in the parentheses, x2. Thus, you get:

(x2)4 = (x2)(x2)(x2)(x2) = (xx)(xx)(xx)(xx) = xxxxxxxx  = x8

Note that x8 also equals x( 2×4 ). This demonstrates the second exponent rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power. Try several other examples to convince yourself. This can be written in general terms as follows: ( xm ) n = x m n

Case 3: Dividing numbers with exponents to a power

Example: Simplify [pic]

In this case, rewriting the numerator and denominator in expanded form and canceling out common factors (which is done in simplifying fractions) you get:

[pic]

Note that x[pic] also equals x[pic]. This demonstrates the third exponent rule: Whenever you divide two terms with the same base, you can simply subtract the denominator exponent from the numerator exponent. Try several other examples to convince yourself. This can be written in general terms as follows: [pic].

Directions: Part II—After reading the informational text on the properties of exponents, work with a partner on the questions below and be ready to discuss your answers with the rest of the class. You must use the informational text to justify your answers.

1. In your own words (using as few words as possible), explain each case presented in the informational text.

Case 1:

Case 2:

Case 3:

2. Determine which case the following problem fits and then simplify using the property.

[pic]

3. Determine which case the following problem fits and then simplify using the property.

[pic]

4. Determine which case the following problem fits and then simplify using the property.

[pic]

5. Challenge: The following problem utilizes all three of the properties, Using what you learned about the properties of exponents, see if you can simplify the following expression and be ready to justify how you got your solution using the properties of exponents.

[pic]

Directions: Part II—After reading the informational text on the properties of exponents, work with a partner on the questions below and be ready to discuss your answers with the rest of the class. You must use the informational text to justify your answers.

1. In your own words (using as few words as possible), explain each case presented in the informational text.

Case 1: See Student Work

Case 2: See Student Work

Case 3: See Student Work

2. Determine which case the following problem fits and then simplify using the property.

[pic] Answer: Case 3; [pic]

3. Determine which case the following problem fits and then simplify using the property.

[pic] Answer: Case 1; [pic]

4. Determine which case the following problem fits and then simplify using the property.

[pic] Answer: Case 2; [pic]

6. Challenge: The following problem utilizes all three of the properties, Using what you learned about the properties of exponents, see if you can simplify the following expression and be ready to justify how you got your solution using the properties of exponents.

[pic] Answer: [pic][pic] or [pic]

Directions: Take a look at the chart below and determine your own understanding of the terms shown below. If you feel like you fully understand the term and its use in math, put a check mark in the + column and then give an example and your own definition of the term in the last two columns. If you don’t understand a math term, put a check mark in the – column. Finally, if you feel you have some understanding but are not completely comfortable with the math term, put a check mark in the ? column and then give an example and definition based upon what your understanding is at this point.

|Math |+ |? |- |Example |Definition of Math Term |

|Term |(understand concept |(understand concept to |(don’t understand | | |

| |completely) |some degree but still |concept) | | |

| | |have questions) | | | |

|Factor | | | | | |

| | | | | | |

|Coefficient | | | | | |

| | | | | | |

|Term | | | | | |

| | | | | | |

Use the following expression to answer the questions below: 3x² + 4x + 2

1. How many terms are in the expression? Identify them.

2. What is the coefficient of each term?

3. Pick one of the terms in the expression and identify the factors of the term.

Directions: Take a look at the chart below and determine your own understanding of the terms shown below. If you feel like you fully understand the term and its use in math, put a check mark in the + column and then give an example and your own definition of the term in the last two columns. If you don’t understand a math term, put a check mark in the – column. Finally, if you feel you have some understanding but are not completely comfortable with the math term, put a check mark in the ? column and then give an example and definition based upon what your understanding is at this point.

|Math |+ |? |- |Example |Definition of Math Term |

|Term |(understand concept |(understand concept to |(don’t understand | | |

| |completely) |some degree but still |concept) | | |

| | |have questions) | | | |

|Factor | | | | | |

| | | | | | |

|Coefficient | | | | | |

| | | | | | |

|Term | | | | | |

| | | | | | |

*See Student Work on Vocabulary Self-Awareness Chart Above

Use the following expression to answer the questions below: 3x² + 4x + 2

1. How many terms are in the expression? Identify them.

Answer: There are 3 terms in the expression.

Term 1: 3x² Term 2: 4x Term 3: 2

2. What is the coefficient of each term?

Answer: The coefficients are as follows:

Term 1: 3 Term 2: 4 Term 3: 2

Teacher note: Technically, the constant term (term 3) is the “constant coefficient” of the x[pic]term. Some books don’t acknowledge this term as having a coefficient and just say it is a constant (which it is the constant term), but this is actually the coefficient of the zero degree term.

3. Pick one of the terms in the expression and identify the factors of the term.

Possible Answer: Term 2 has factors of 4 and x.

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Algebra I–Part 2

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