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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Roots

If a square has a side of length 7 feet, then it has an area of 72, or 49 square feet. If a square has an area of 36 square feet, then the length of its side is 6 feet. If we know the length of a side, then we square it to find the area. If we know the area, then we must undo the process of squaring to find the length of a side. Undoing the process of squaring is called taking the square root.

Because 32 = 9, 23 = 8, and (-2)4 = 16, we say that 3 is a square root of 9, 2 is the cube root of 8, and -2 is a fourth root of 16. In general, undoing an nth power is referred to as taking an nth root.

 

nth Roots

The number b is an nth root of a if bn = a.

 

Both 3 and -3 are square roots of 9 because 32 = 9 and (-3)2 = 9. Because 24 = 16 and (-2)4 = 16, there are two real fourth roots of 16: 2 and -2. If n is a positive even integer and a is any positive real number, then there are two real nth roots of a. We call these roots even roots. The positive even root of a positive number is called the principal root. The principal square root of 9 is 3, and the principal fourth root of 16 is 2. When n is even, the exponent 1/n is used to indicate the principal nth root. The principal nth root can also be indicated by the radical symbol .

 

Exponent 1/n When n Is Even

If n is a positive even integer and a is a positive real number, then a1/n denotes the positive real nth root of a and is called the principal nth root of a.

 

Caution

An exponent of n indicates nth power, an exponent of -n indicates the reciprocal of the nth power, and an exponent of 1/n indicates nth root.

We will see later that choosing 1/n to indicate nth root fits in nicely with the rules of exponents that we have already studied.

 

Example 1

Finding even roots

Evaluate each expression.

a) 41/2

b) 161/4

c) -811/4

d)

Solution

a) Because 22 = 4, we have 41/2 = 2. Note that 41/2 -2.

b) Because 24 = 16, we have 161/4 = 2. Note that 161/4 -2.

c) Following the accepted order of operations from Chapter 1, we find the root first and then take the opposite of it. Because 34 = 81, we have 811/4 = 3 and -811/4 = -3.

d) Because , we have .

Note that 23 = 8 but (-2)3 = -8. The cube root of 8 is 2, and the cube root of -8 is -2. If n is a positive odd integer and a is any real number, then there is only one real nth root of a. We call this root an odd root.

 

Exponent 1/n When n Is Odd

If n is a positive odd integer and a is any real number, then a1/n denotes the real nth root of a.

 

Example 2

Finding odd roots

Evaluate each expression.

a) 81/3

b) (-27)1/3

c) -321/5

Solution

a) Because 23 = 8, we have 81/3 = 2.

b) Because (-3)3 = -27, we have (-27)1/3 = -3.

c) Because 25 = 32, we have -321/5 = -2.

We do not allow 0 as the base when we use negative exponents because division by zero is undefined. However, positive powers of zero are defined, and so are roots of zero; for example, 04 = 0, and so 01/4 = 0.

 

nth Root of Zero

If n is a positive integer, then 01/n = 0.

 

Caution

An expression such as (-9)1/2 is not included in the definition of roots because there is no real number whose square is -9. The definition of roots does not include an even root of any negative number because no even power of a real number is negative.

The expression 31/2 represents the unique positive real number whose square is 3. Because there is no rational number that has a square equal to 3, the number 31/2 is an irrational number. If we use a calculator, we find that 31/2 is approximately equal to the rational number 1.732. Because the square root of 3 is not a rational number, the simplest representation for the exact value of the square root of 3 is 31/2.