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Undefined Rational Expressions
Writing Equations for Lines Using Sequences
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Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
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Powers and Exponents


Learn the idea of exponentiation and the notion of raising a number to a power.

Exponential notation is very important in all future uses of mathematics.

Raising Numbers to a Power

Remember that the area of a rectangle is the product of the lengths of the sides. When the rectangle is a square, we can find the area by multiplying one side times itself. In the same way, the volume of a cube is found by multiplying one side times itself three times.

The idea of multiplying a number repeatedly by itself comes up so often that a special notation has been invented for it.


We will write x n for the result of multiplying n "repeats" of x together. The number x is called the base, and the number n is called the exponent.

For instance,

base = x, exponent = 4
base = 2, exponent = 6
base = a, exponent = 3

For the expression x 2, we say x squared. For the expression x 3, we say x cubed. Otherwise, for x n, we say x raised to the nth power. For example, 5 raised to the 4th power is written 5 4, and a raised to the fourth power is written a 4. Note that the exponent can be 1. For example, x 1 = x . A power of a number a is a number found by raising a to a power.

Let's see some examples of expanding powers into products and raising numbers to exponents.

Example 1

Write 3 5 , x 7 , a 4 , and b 8 as products.


3 5 = 3 · 3 · 3 · 3 · 3

x 7 = x · x · x · x · x · x · x

a 4 = a · a · a · a

b 8 = b · b · b · b · b · b · b · b


Example 2

Write 4 · 4 · 4, d · d · d · d · d · d , and ( z + y ) · ( z + y ) using exponents.


4 · 4 · 4 = 4 3

d · d · d · d · d · d = d 6

( z + y ) · ( z + y ) = ( z + y ) 2

Now is a good time to explain the roles of the powers of ten in our place-value system. That is, every number can be written as a sum of single-digit number times a power of ten. When the powers of ten are written explicitly, we call the expression the expanded form of the number. The expanded form of 1234 is shown below. Notice that 1 = 10 0. Any number raised to the zero power equals 1.

We have introduced the idea of exponentiation. Now, it is important to extend the rules for order of operations to include exponents.

Definition of Order of Operations

1. Do all operations within parentheses first; start with an innermost pai r of parentheses.

2. Evaluate all powers in order from left to right.

3. Perform all multiplications and divisions in order from left toright.

4. Perform all additions and subtractions in order from left to right.

Here are some examples to illustrate this idea.


Example 3

Evaluate ab 2 if a = 2 and b = 3.


ab 2 = 2 · 3 2 Replace a with 2 and b with 3.
  = 2 · 9 Evaluate the power.
  = 18 Multiply 2 and 9.

Notice that we evaluated the power before the product. If we hadn't done this, we would have multiplied first to get 6 2 or 36.


Example 4

Evaluate (2 + 4 2 ) (9 2 + 3 · 4).


(2 + 4 2 ) · (9 2 + 3 · 4) = ( 2 + 16 ) · (81 + 3 · 4 ) Evaluate the powers.
  = ( 2 + 16 ) · (81 + 12 ) Multiply 3 and 4.
  = 18 · 93 Add 2 and 16 and81 and 12.
  = 1674  

Notice the order of operations. If the operations were performed in any other order, we would have arrived at a different, incorrect answer.

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