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.
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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.
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.
Simplify each expression.
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.
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.
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)
Using Inequality SymbolsEXAMPLE 4
Reverse symbol.Write statements that change the direction of the inequality symbols.
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