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Module 6 Test Review

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**Now is a chance to review all of the great stuff you have been learning in Module 6!**

Numerical Expressions Algebraic Expressions Parts of an Expression Evaluating Algebraic expressions Algebraic Properties Equivalent Expressions

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Exponential form Exponential form is just a simplified way of writing a multiplication expression where a number is being multiplied by itself

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**Area in Exponential Form**

Since the 5 is being multiplied by itself 2 times, you can use an exponent of 2. The area 5 ft × 5 ft written in exponential form is 52 ft2. When the exponent is a 2, this is called squaring the base. So you can say "five squared."

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**Volume in Exponential Form**

To calculate the volume of the circus cube you would multiply 5 ft × 5 ft × 5 ft. 5 is the base, but this time it is multiplied 3 times so the exponent in this case is 3. Therefore, the exponential form of the volume is 53 ft3. When an exponent is a 3, this is called cubing the base. So you can say "five cubed."

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**Simplifying exponential numbers**

35 = 3 x 3 x 3 x 3 x 3 = 3 x 3 x 3 x 3 x 3 9 x 3 x 3 x 3 27 x 3 x 3 Caution 81 x 3 When simplifying an exponent, you must remember that 73 = 7 × 7 × 7. It does not equal 7 × 3 or 7·3 or 73 243

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**Key Terms Verbal expression:**

A mathematical expression in which all mathematical operations are written using words; also called a mathematical phrase; an example is the product of three and a number. Variable: A letter that holds the place for some unknown value in an algebraic expression, such as x or y. Algebraic expression: An expression that contains numbers, variables, and mathematical operations like addition, subtraction, etc; an example is x + 2

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Verbal Phrases This chart groups commonly used phrases by their meaning

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**Translating Expressions**

How do you translate the expression: “Double the total number of brownies and cookies?”

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Example Sam has to pay a $5 admissions fee for each person who attends the party. Sam’s expression now contains two operations. The verbal expression for the cost of admissions is “the product of five and the quantity forty minus a number.” The first operation is to determine the total number of friends. The second operation is to multiply by five. Adding the parentheses around the first operation will ensure that Sam subtracts before he multiplies. Sam can write the algebraic expression as:

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Tricks of Translating An algebraic expression uses numbers, letters, and symbols instead of words. All you will need to do is change the words into math. Since algebraic expressions contain variables like the letter x, you should no longer use x to represent multiplication

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Tricks of Translating It is important to pay attention to the order of quantities when translating When the phrases “less than” and “subtracted from” are translated to an algebraic expression, the order of the numbers must be reversed.

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**Translating expressions**

Sam invited 40 friends, but his mom mentioned that some might not come. This means he would have “forty minus some number.” First, identify the action phrase in the verbal expression to know what operation is being used. In this expression, the action phrase is “minus,” which means to subtract. So the verbal expression written in algebraic form is:

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Example Rewrite “the quotient of eight and an unknown value” into an algebraic expression In this expression, the action phrase is "quotient", meaning division.

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**Parts of an Expression Example: 2x2 – 3y – z + 6 Variable:**

Expressions, whether they are mathematical or verbal, can contain variables. A variable is a letter that holds the place for some unknown value in an expression. expressions can have two or more unknown values. In those cases, you simply use a new letter for each unknown value. the given mathematical expression has three unknown values because there are three variables: x, y, and z Term: Expressions are made up of terms separated by a plus or minus sign. Terms can contain variables, numbers, or products of variables and numbers. The given expression contains four total terms: 2x2, −3y, −z, and 6. Notice how the plus or minus sign is attached to the term immediately following it.

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**Parts of an Expression Example: 2x2 – 3y – z + 6 Factor:**

Factors are numbers you multiply together to produce a product. Look at the factors that are easily seen in the given expression. 2 and x2 are factors of 2x2 −3 and y are factors of −3y −1 and z are factors of −z Coefficient: When multiplying a variable and a number to write a term, the number is listed first and is called the coefficient. It is important to remember the sign in front of the term also goes with that term. The coefficients in this expression are 2, −3, and −1

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**Parts of an Expression Example: 2x2 – 3y – z + 6 Constant:**

Constants are numbers that stand alone. They are called constants because they have fixed value The constant of this expression is 6

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**Identify the parts of: 4(t + 3) – s**

Example Identify the parts of: 4(t + 3) – s Variables: t and s Term: There are two terms in this expression: 4(t + 3) and –s Factor: Parentheses not only group operations together, they also mean multiplication of two factors. 4 and t + 3 are factors of the term 4(t + 3). As well, –1 and s are factors of the terms –s Coefficient: The coefficients in this expression are 4, 1, and –1 Constant: In this expression, the second factor of the term 4(t + 3) has a constant of 3 Remember, if there is no number in front of a variable, it is understood as 1 times that variable and is not written

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**Evaluating Expressions**

Substitution means replacing one thing with something else You can substitute for the variables when you know their values. Then you follow the order of operations to simplify the expression

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**Evaluating Expressions**

Let’s use substitution to evaluate the expression 5x + 100, if x equals 7 Since you know the value of the variable is 7, you can replace the letter x with the number 7. That’s all there is to substitution The algebraic expression is now a numerical expression and can be easily simplified. To evaluate the expression, follow the order of operations.

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Example 2 Evaluate the expression 2x2 + 4y, when x = 8 and y = 1.5

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Commutative Property The Commutative Property of Addition says that the order of the addends in a sum does not matter. The Commutative Property of Multiplication says that the order of the factors in a product does not matter.

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**Commutative Property – Examples**

Sarah drove 30 miles to the beach and m extra miles to the store. The total distance can be represented as 30 + m or m + 30. The sum is the same either way because of the commutative property The total amount of gear needed for the beach day can be represented as 5 + (x + 2). You can see it as the sum of 5 and another quantity x + 2. According to the commutative property of addition, you can change the order around of the two addends. 5 + (x + 2) = (x + 2) + 5 The sum is the same either way because of the commutative property of addition.

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Associative Property The Associative Property of Addition says that the grouping of the addends in a sum does not matter. The Associative Property of Multiplication says that the grouping of the factors in a product does not matter.

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**Associative Property - examples**

Sarah swam 30 minutes in the morning, 45 minutes at lunch, and x minutes in the afternoon. The total amount she swam can be represented with the expression 30 + (45 + x) or ( ) + x because of the associative property. Sarah bought 2 cans of sunscreen for some friends, f, and each can cost $4. Sarah can represent the total cost of the sunscreen as (f ⋅ 2) ⋅ 4 or f ⋅ (2 ⋅ 4) because of the associative property.

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Identity Property The Identity Property of Addition states that adding 0 to a number does not change the identity (or value) of the number. The Identity Property of Multiplication states that multiplying a number by 1 does not change the identity (or value) of the number Zero is the identity element for addition because zero has no effect on the value in a sum. One is the identity element for multiplication because it has no effect on the value in a product. Because of the identity properties, you can manipulate the expression to suit your needs while maintaining equality between two expressions.

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**Caution with the properties**

The commutative property does not work for subtraction or division. This means you cannot reverse the order of a subtraction or division expression and keep the same value. The associative property does not work if the expression contains more than one operation The associative property does not work if the expression contains subtraction or division. This means you cannot move the parentheses around on a subtraction or division expression and always keep the same value

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**Distributive Property**

Rule Distributive Property (with a sum): a(b + c) = a ⋅ b + a ⋅ c Distributive Property (with a difference): a(b − c) = a ⋅ b − a ⋅ c The distributive property says that any number multiplied to a sum or difference of two or more numbers is equal to the sum or difference of the products. The property allows you to rewrite a product as a sum or difference to suit your needs without changing the value of the expression

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**Distributive Property - example**

Apply the distributive property to generate an equivalent expression for 4(x + 10) Step 1: Multiply the first term in the parentheses by the factor. In this example, remember there is an invisible 1 as the coefficient for the x variable. Step 2: Bring over the mathematical operation in the parentheses. Step 3: Multiply the second term in the parentheses by the factor, and simplify. Therefore, 4(x + 10) is equivalent to 4x + 40 because of the distributive property.

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Key Terms Like terms: Terms in an expression that have the same variable part; for example, 5x and 4x are like terms. Unlike terms: Terms in an expression that do not have the same variable part; for example, 5x and y are unlike terms.

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**Examples of like and unlike terms**

7x, x, and –3x are like terms because all the terms contain the same variable part x. y2, 4y2, and –8y2 are like terms because all the terms contain the same variable part y2. 4ab, –8ab, and ab are like terms because all the terms contain the same variable part ab. All constants terms are like terms because they do not contain a variable 7x and y both have a single variable, but the terms are not alike since different variables are used. 4a and 4b, although they have the same coefficient are unlike terms because the terms do not have the same variable. 2x and –5x2, although each term contains the same variable x, are not alike because each x variable has a different exponent. 6x and 4xy, although both terms have an x variable, only one term has the y variable, so these terms are not like terms.

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**Combining Like Terms Combine like terms:**

A process of combining terms that have identical variable parts. Combining like terms means combining the coefficients of the like terms. The purpose of combining like terms is to help simplify an expression.

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**Combining Like Terms – example**

In the expression 3y + 2y, notice how both terms contain the exact same variable y. For this expression, you can combine the like terms by adding the coefficients of the terms. Use the distributive property to factor out the y variable, and then add the coefficients to combine the like terms. Therefore, the expression 3y + 2y simplifies to 5y because of combining like terms using the distributive property. This means that 3y + 2y, (3 + 2)y, and 5y are all equivalent expressions.

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**Equivalent Expressions**

In an algebraic expression, you can use the mathematical properties to group and combine like terms. The properties also allow you to prove that two expressions are equivalent.

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**Proving Equivalent Expressions: Method 1**

Show that the expression 3x x + 5 is equivalent to 5x + 7. Can you identify which properties are used to show that the two expressions are equivalent? Think about which property allows you to move from step to step

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**Proving Equivalent Expressions: Method 2**

Show that the expression 3x x + 5 is equivalent to 5x + 7. Can you identify which properties are used to show that the two expressions are equivalent? Think about which property allows you to move from step to step You can prove these expressions are equivalent by using substitution. Substitute 1 for x, and simplify. This is a great way to check your math work to ensure you used the properties correctly.

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**Proving Equivalent Expressions: Method 3**

Show that the expression 3x x + 5 is equivalent to 5x + 7. Can you identify which properties are used to show that the two expressions are equivalent? Think about which property allows you to move from step to step If two expressions are truly equivalent, the value of the expression should be equal regardless of the number that is substituted. Let’s substitute 5 for x to be sure this works for another number.

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**Numerical Expressions Algebraic Expressions Parts of an Expression **

You have now had a chance to review all of the great stuff you learned in Module 6! Numerical Expressions Algebraic Expressions Parts of an Expression Evaluating Algebraic expressions Algebraic Properties Equivalent Expressions Have you completed all assessments in module 6? Have you completed your Module 6 DBA? Now you are ready to move forward and complete your module 6 test. Please make sure you are ready to complete your test before you enter the test session.

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