CHAPTER 1 REVIEW Section 1 - Algebraic Expressions
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1 CHAPTER 1 REVIEW Section 1 - Algebraic Expressions A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable. The terms of an expression are the parts of a mathematical expression that are separated by a plus (+) or minus ( ) sign. Each term is either a number or the product of a number (sometimes an understood 1) and one or more variables. For Example, 6x + 4y 2x + 2 The four terms of the above expression are 6x, 4y, 2x and 2. The coefficient is the numerical portion of the term (the number next to the variable) A constant is an isolated number (a number not attached to a variable. Example 1: List the variables, coefficients and constants in the expression below. 6x + 4y 2x + 3 Variables x, y Coefficients 6, 4, 2 Constant 3 Evaluating Expressions Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. To evaluate an algebraic expression, replace the variable or variables with known values and then use the order of operations. For your convenience, you can use parenthesis when you plug in negative values to the variables.
2 Example 1: Evaluate the expression. Use the values p = 4, and n = 14. 3( p + n ) To evaluate the expression, first substitute 4 for p and 14 for n in the expression. 3( p + n ) = 3(( 4) + 14) Use the order of operations to simplify. First work out inside the parenthesis. Add the numbers 4 and 14. = 3(10) Now, multiply 3 and 10. 3(10) = 30 So, when p is 4 and n is 14, the value of the given expression is 30. Example 2: Evaluate each expression if x = 7, and y = 3. To evaluate the expression, replace x with 7 and y with 3. Use the order of operations and simplify the expression. Perform multiplication and division from left to right. Add 7 and 2. =9 Therefore, for the given values of x and y, the value of the expression is 9.
3 Exponents There's one more operation besides addition, subtraction, multiplication, and division. That's exponentiation, notated by superscript (e.g. x 2 or e x ), or sometimes -- especially on calculators and computers -- using the symbol ^. 6 2 means "6 to the power 2", or in other words: 6 2 = 6 6 = 36. (Remember, it's not just 6 2.) The 6 is the base and the 2 is the exponent.. The exponent tells you how many times to use the base as a factor. In the example below, the base is 3 and the exponent is 4, so we find the product of four threes. 3 4 = = 9 9 = 81. Example 1: Write the algebraic expression using exponents. 4 x t x t x t x t x t x t x t The factor t appears 7 times, so the exponent is 7 and the base is t. 4t 7
4 Section 1.2 Writing Expressions Below are a few words and phrases to look out for when translating from English to mathematical symbols. Pay attention to the order of the terms! ENGLISH 3 added to a number x MATH SYMBOLS 3 plus a number x 3 more than a number x 3 greater than a number x 3 increased by a number x 3 + x the total of 3 and a number x the sum of 3 and a number x a number x minus 5 5 less than a number x 5 fewer than a number x 5 subtracted from a number x 5 a number x decreased by 5 the difference of a number x and 5 twice a number x 6 times as many as a number x 6 multiplied by a number x the product of 6 and x 2 x 6 x the quotient of a number n and 8 a number n divided by 8 the square of a number x 2 the cube of a number x 3
5 is/is equal to/is the same as = is less than < is greater than/is more than > is at most / is no more than is at least / is no less than Also, the following four definitions are important: Sum Difference Product Quotient Answer to an addition problem Answer to a subtraction problem Answer to a multiplication problem Answer to a division problem
6 Section 1.3 Properties of Addition & Multiplication The Commutative Properties of Addition & Multiplication The Commutative Properties The commutative properties state that the order in which you add or multiply two real numbers does not affect the result. The Commutative Property of Addition: a + b = b + a = = ( 3) = ( 3) + 20 = 17 The Commutative Property of Multiplication: ab = ba 4 5 = 5 4 = 20 ( 2)(8) = (8)( 2) = 16 The Associative Properties of Addition & Multiplication The Associative Laws (or the Associative Properties) The associative properties state that when you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result. The Associative Property of Addition: (a + b) + c = a + (b + c) (2 + 3) + 5 = = (3 + 5) = = 10
7 The Associative Property of Multiplication: (ab)c = a(bc) (5 7) 6 = 35 6 = (6 7) = 5 42 = 210 Addition Property of Zero The sum of any number and 0 is that number a + 0 = a = 2.15 Multiplication Property of Zero The product of any number and 0 is zero. a x 0 = 0 3 x 0 = 0 Multiplication Property of One The product of any number and 1 is that number. a x 1 = a 7 x 1 = 7
8 Seciont The Distributive Property The Distributive Property states that, for all real numbers x, y, and z, x(y + z) = xy + xz. Take 3(6 + 7). We can add within the parentheses, and then multiply: 3(6 + 7) = 3(13) = 39 Or, you can multiply each addend by the 3, and then add. 3(6 + 7) = 3(6) + 3(7) = = 39 Either way, you get the same answer. Using the Distributive Property with Variables You can use the Distributive Property to simplify algebraic expressions. 7p + 3q 21p + 8q = (7 21)p + (3 + 8)q = 14p + 11q Using the Distributive Property to do Mental Multiplication You can sometimes use the Distributive Property to break difficult multiplication problems into two or more easy ones that you can do in your head. Example 1: = 7(1000 3) = 7(1000) 7(3) = = 6979 Example 2: = ( )3 = 1000(3) + 300(3) + 9(3) = = 3927
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