3 Like terms contain the same variables with the same exponents. What are like terms?Like terms contain the same variables with the same exponents.
4 Example 1 a) Are like terms? Yes. Each variable contains the variable p with and exponent of 2. They are -terms.b) Are like terms?No. Although each contains the variable f, the exponents are not the same.
5 Combine like termsWe combine like terms using the distributive property. We can add and subtract only those terms that are like terms.
6 Example 2 Combine like terms. a) b) We combine the coefficients of the like terms.b)
7 Parentheses in an Expression If an expression contains parentheses, use the distributive property to clear the parentheses, and then combine like terms.
30 Integer Exponents with Variable Bases Section 2.3bInteger Exponents with Variable Bases
31 Expressions Containing Variable Bases The rules that apply to real number bases also apply when the bases are variables.
32 Example 1 Evaluate. Assume the variable does not equal zero. a) b) Solutiona)b)
33 Example 2Rewrite the expression with positive exponents. Assume the variable does not equal zero.a) b)Solutiona)b)
34 Definition: If m and n are any integers and and are real numbers not equal to zero, thenTo rewrite the original expression with only positive exponents, the terms with the negative exponents “switch” their positions in the fraction.
35 Example 3Rewrite the expression with positive exponents. Assume the variables do not equal zero.a) b)Solutiona)b)The exponent on t is positive, so do not change its position in the expression.
37 Quotient Rule for Exponents Quotient Rule for Exponents: If m and n are anyintegers and thenTo apply the quotient rule, the bases must be the same. Subtract the exponent of the denominator from the exponent of the numerator.
38 Example 1 Simplify. Assume the variable does not equal zero. a) b) c) Solutiona) b)c)
39 Example 2Simplify the expression. Assume the variables do not equal zero.Solution
40 Putting the Rules Together Mid-Chapter SummaryPutting the Rules Together
41 Example 1 Simplify . Assume the variables do not equal zero. SolutionBegin by taking the reciprocal of the base to eliminate the negative on the exponent on the outside of the parentheses.
43 Definition:A number is in scientific notation if it is written in the form where and n is an integer.means that is a number that has one nonzero digit to the left of the decimal point. Here are two numbers in scientific notation:
44 Example 1 Write without exponents a) b) Solution a) Move the decimal point 4 places to the right. Multiplying by a positive power of 10 will make the result larger thanb) Move the decimal point 2 places to the left. Multiplying 1.07 by a negative power of 10 will make the result smaller than 1.07.
45 Example 2 Write each number in scientific notation. a) b) Solution a) To write in scientific notation, the decimal point must go between the 5 and the 2. This will move the decimal point 6 places.b) To write in scientific notation, the decimal point must go after the 9. This will move the decimal point 5 places.