Chapter 1. Section 2. Use exponents. Use the rules for order of operations. Use more than one grouping symbol. Know the meanings of ≠, < , > , ≤ , and ≥ . Translate word statements to symbols. Write statements that change the direction of inequality symbols.
Use the rules for order of operations.
Use more than one grouping symbol.
Know the meanings of ≠, <, >, ≤, and ≥.
Translate word statements to symbols.
Write statements that change the direction of inequality symbols.
Repeated factors are written with an exponent. For example, in the prime factored form of 81, written , the factor 3 appears four times, so the product is written as 34, read “3 to the fourth power.”Use exponents.
For this exponential expression, 3 is the base, and 4 is the exponent, or power.
A number raised to the first power is simply that number.
Squaring, or raising a number to the second power, is NOT the same as doubling the number.
Evaluating Exponential ExpressionsEXAMPLE 1
= 9 • 9
Many problems involve more than one operation. To indicate the order in which the operations should be performed, we often use grouping symbols.
Consider the expression .
If the multiplication is to be performed first, it can be written , which equals , or 11.
If the addition is to be performed first, it can be written , which equals , or 21.Use the rules for order of operations.
For example, in , the expression is considered to be grouped in the numerator.
To work problems with more than one operation, we use the following order of operations.Use the rules for order of operations. (cont’d)
If grouping symbols are present,simplify within them, innermost first (and above and below fraction bars separately), in the following order:
Step 1:Apply allexponents.
Step 2:Do anymultiplicationsordivisionsin the order in which they occur, working from left to right.
Step 3:Do anyadditionsorsubtractionsin the order in which they occur, working from left to right.
If no grouping symbols are present, start with Step 1.
Use the memory device “PleaseExcuseMyDearAuntSally” to help remember the rules for order of operations: Parentheses, Exponents, Multiply, Divide, Add, Subtract.
Using the Rules for Order of OperationsEXAMPLE 2
In expressions such as 3(7) or (─5)(─4),multiplication is understood.
Simplify each expression.
An expression with double (or bars.nested) parentheses, such
as , can be confusing. For clarity, we often use brackets , [ ], in place of one pair of parentheses.
The expression can be written as the quotient below, which shows
that the fraction bar “groups” the numerator and denominator separately.Use more than one grouping symbol.
Know the meanings of ≠, bars.<, >, ≤, and ≥.Objective 4
The symbols ≠, bars., , ≤, and ≥ are used to express an inequality, a statement that two expressions may not be equal. The equality symbol (=) with a slash though it means “is not equal to.”
For example, 7 is not equal to 8.
The symbol represents “is less than,” so
7 is less than 8.
The symbol means “is greater than.” For example
8 is greater than 2.Know the meanings of ≠, <, >, ≤, and ≥.
Remember that the “arrowhead” always points to the lesser number.
Two other symbols, ≤ bars. and ≥, also represent the idea of inequality. The symbol ≤ means “less than or equal to,” so
5 is less than or equal to 9.
Note: If either the part or the = part is true, then the inequality ≤ is true.
The ≥ means “is greater than or equal to.” Again
9 is greater than or equal to 5.
Know the meanings of ≠, <, >, ≤, and ≥. (cont’d)
Determine whether each statement is bars.true or false.
Using Inequality SymbolsEXAMPLE 4
Nine is equal to eleven minus two.
Fourteen is greater than twelve.
Two is greater than or equal to two.
Translating from Words to SymbolsEXAMPLE 5
Any statement with symbols. can be converted to one with >, and any statement with > can be converted to one with . We do this by reversing the order of the numbers and the direction of the symbol.
Reverse symbol.Write statements that change the direction of the inequality symbols.
Write the statement as another true statement with the inequality symbol reversed.
Converting between Inequality SymbolsEXAMPLE 6
Equality and inequality symbols are used to write mathematical sentences,while operations symbols (+, -, ·, and ÷) are used to write mathematical expressions. Compare the following:
Sentence: 4 10 gives a relationship between 4 and 10
Expression: 4 + 10 tells how to operate on 4 and 10 to get 14