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**Variables and Exponents**

Chapter 2 Variables and Exponents

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**Simplifying Expressions**

Section 2.1 Simplifying Expressions

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**Like terms contain the same variables with the same exponents.**

What are like terms? Like terms contain the same variables with the same exponents.

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**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.

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Combine like terms We combine like terms using the distributive property. We can add and subtract only those terms that are like terms.

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**Example 2 Combine like terms. a) b)**

We combine the coefficients of the like terms. b)

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**Parentheses in an Expression**

If an expression contains parentheses, use the distributive property to clear the parentheses, and then combine like terms.

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Example 3 Combine like terms and simplify.

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**Translate English Expressions to Mathematical Expressions**

Read the phrase carefully, choose a variable to represent the unknown quantity, then translate the phrase to a mathematical expression.

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Example 4 Write a mathematical expression for nine less than twice a number. Let x = the number 2x – 9 The expression is 2x – 9.

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**The Product Rule and Power Rules of Exponents**

Section 2.2a The Product Rule and Power Rules of Exponents

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**Definition: An exponential expression of the form is where is any real**

number and n is a positive integer. The base is and n is the exponent.

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Example 1 Identify the base and the exponent in each expression and evaluate. a) is the base, and 4 is the exponent. b) is the base, and 4 is the exponent. c) is the base, and 4 is the exponent.

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**Product Rule for Exponents**

Product Rule: Let be any real number and let m and n be positive integers. Then,

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Example 2 Find each product. a) b) Solution a) b)

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Basic Power Rule Basic Power Rule: Let be any real number and let m and n be positive integers. Then,

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Example 3 Simplify using the power rule. a) b) Solution a) b)

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**Power Rule for a Product**

Power Rule for a Product: Let and be any real numbers and let n be a positive integer. Then,

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Example 4 Simplify each expression. a) b) Solution a) b)

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**Power Rule for a Quotient**

Power Rule for a Quotient: Let and be any real numbers and let n be a positive integer. Then,

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Example 5 Simplify using the power rule for quotients. a) b) Solution

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**Combining the Rules of Exponents**

Section 2.2b Combining the Rules of Exponents

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**Combining the Rules of Exponents**

When we combine the rules of exponents, we follow the order of operations.

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Example 1 Simplify. a) b) Solution a) b)

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**Integer Exponents with Real Number Exponents**

Section 2.3a Integer Exponents with Real Number Exponents

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Definition: Zero as an Exponent: If then

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Example 1 Evaluate. a) b) Solution a) b)

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**Definition: Negative Exponent: If n is any integer and then**

To rewrite an expression of the form with a positive exponent, take the reciprocal of the base and make the exponent positive.

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Example 2 Evaluate. a) b) Solution a) b)

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**Integer Exponents with Variable Bases**

Section 2.3b Integer Exponents with Variable Bases

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**Expressions Containing Variable Bases**

The rules that apply to real number bases also apply when the bases are variables.

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**Example 1 Evaluate. Assume the variable does not equal zero. a) b)**

Solution a) b)

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Example 2 Rewrite the expression with positive exponents. Assume the variable does not equal zero. a) b) Solution a) b)

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**Definition: If m and n are any integers and and**

are real numbers not equal to zero, then To rewrite the original expression with only positive exponents, the terms with the negative exponents “switch” their positions in the fraction.

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Example 3 Rewrite the expression with positive exponents. Assume the variables do not equal zero. a) b) Solution a) b) The exponent on t is positive, so do not change its position in the expression.

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Section 2.4 The Quotient Rule

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**Quotient Rule for Exponents**

Quotient Rule for Exponents: If m and n are any integers and then To apply the quotient rule, the bases must be the same. Subtract the exponent of the denominator from the exponent of the numerator.

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**Example 1 Simplify. Assume the variable does not equal zero. a) b) c)**

Solution a) b) c)

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Example 2 Simplify the expression. Assume the variables do not equal zero. Solution

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**Putting the Rules Together**

Mid-Chapter Summary Putting the Rules Together

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**Example 1 Simplify . Assume the variables do not equal zero.**

Solution Begin by taking the reciprocal of the base to eliminate the negative on the exponent on the outside of the parentheses.

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Section 2.5 Scientific Notation

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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:

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**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 than b) 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.

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**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.

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