# Variables and Exponents

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Variables and Exponents
Chapter 2 Variables and Exponents

Simplifying Expressions
Section 2.1 Simplifying Expressions

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

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.

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

Example 2 Combine like terms. a) b)
We combine the coefficients of the like terms. b)

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

Example 3 Combine like terms and simplify.

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.

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

The Product Rule and Power Rules of Exponents
Section 2.2a The Product Rule and Power Rules of Exponents

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.

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.

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

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

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

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

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

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

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

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

Combining the Rules of Exponents
Section 2.2b Combining the Rules of Exponents

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

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

Integer Exponents with Real Number Exponents
Section 2.3a Integer Exponents with Real Number Exponents

Definition: Zero as an Exponent: If then

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

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.

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

Integer Exponents with Variable Bases
Section 2.3b Integer Exponents with Variable Bases

Expressions Containing Variable Bases
The rules that apply to real number bases also apply when the bases are variables.

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

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

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.

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.

Section 2.4 The Quotient Rule

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.

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

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

Putting the Rules Together
Mid-Chapter Summary Putting the Rules Together

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.

Section 2.5 Scientific Notation

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:

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.

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.