Exponents and Radicals

Exponents and Radicals
10 Exponents and Radicals 10.1 Radical Expressions and Functions 10.2 Rational Numbers as Exponents 10.3 Multiplying Radical Expressions 10.4 Dividing Radical Expressions 10.5 Expressions Containing Several Radical Terms 10.6 Solving Radical Equations 10.7 The Distance and Midpoint Formulas and Other Applications The Complex Numbers Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Radical Expressions and Functions
10.1 Radical Expressions and Functions Square Roots and Square-Root Functions Expressions of the Form Cube Roots Odd and Even nth Roots Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Square Roots and Square-Root Functions
When a number is multiplied by itself, we say that the number is squared. Often we need to know what number was squared in order to produce some value a. If such a number can be found, we call that number a square root of a. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Square Root The number c is a square root of a if c 2 = a.
Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

16 has –4 and 4 as square roots because (–4)2 = 16 and 42 = 16.
For example, 16 has –4 and 4 as square roots because (–4)2 = 16 and 42 = 16. –9 does not have a real-number square root because there is no real number c for which c 2 = –9. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find two square roots of 49.

Whenever we refer to the square root of a number, we mean the nonnegative square root of that number. This is often referred to as the principal square root of the number. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Principal Square Root The principal square root of a nonnegative number is its nonnegative square root. The symbol is called a radical sign and is used to indicate the principal square root of the number over which it appears. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify each of the following.

The expression under the radical sign is called the radicand.
is also read as “the square root of a,” “root a,” or “radical a.” Any expression in which a radical sign appears is called a radical expression. The following are examples of radical expressions: The expression under the radical sign is called the radicand. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Expressions of the Form
It is tempting to write but the next example shows that, as a rule, this is untrue. Example Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplifying For any real number a,
(The principal square root of a2 is the absolute value of a.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Cube Roots We often need to know what number was cubed in order to produce a certain value. When such a number is found, we say that we have found the cube root. For example, 3 is the cube root of 27 because 33 =27. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Odd and Even nth Roots The fourth root of a number a is the number c for which c4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2, we do not write it. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify each expression.

Example Simplify each expression, if possible. Assume that variables can represent any real number. Solution

Simplifying nth Roots n a |a| Even Positive Negative Not a real number
Odd Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rational Numbers as Exponents
10.2 Rational Numbers as Exponents Rational Exponents Negative Rational Exponents Laws of Exponents Simplifying Radical Expressions Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rational Exponents Consider a1/2a1/2. If we still want to add exponents when multiplying, it must follow that a1/2a1/2 = a1/2 + 1/2, or a1. This suggests that a1/2 is a square root of a. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

When a is nonnegative, n can be any natural number greater than 1
When a is nonnegative, n can be any natural number greater than 1. When a is negative, n can be any odd natural number greater than 1. Note that the denominator of the exponent becomes the index and the base becomes the radicand. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Positive Rational Exponents
For any natural numbers m and n (n not 1) and any real number a for which exists, Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Write an equivalent expression using exponential notation.

Negative Rational Exponents
For any rational number m/n and any nonzero real number a for which exists, Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Laws of Exponents For any real numbers a and b and any rational exponents m and n for which am, an, and bm are defined: 1. 2. 3. 4. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use the laws of exponents to simplify

Simplifying Radical Expressions
Many radical expressions contain radicands or factors of radicands that are powers. When these powers and the index share a common factor, rational exponents can be used to simplify the expression. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Simplify Radical Expressions
1. Convert radical expressions to exponential expressions. 2. Use arithmetic and the laws of exponents to simplify. 3. Convert back to radical notation as needed. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Multiplying Radical Expressions
10.3 Multiplying Radical Expressions Multiplying Radical Expressions Simplifying by Factoring Multiplying and Simplifying Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Multiplying Radical Expressions
Note that This example suggests the following. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Product Rule for Radicals
For any real numbers (The product of two nth roots is the nth root of the product of the two radicands.) Rational exponents can be used to derive this rule: Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplifying by Factoring
The number p is a perfect square if there exists a rational number q for which q2 = p. We say that p is a perfect nth power if qn = p for some rational number q. The product rule allows us to simplify whenever ab contains a factor that is a perfect nth power. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Using The Product Rule to Simplify
( must both be real numbers.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Simplify a Radical Expression with Index n by Factoring
1. Express the radicand as a product in which one factor is the largest perfect nth power possible. 2. Take the nth root of each factor. 3. Simplify the expression containing the perfect nth power. 4. Simplification is complete when no radicand has a factor that is a perfect nth power. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

It is often safe to assume that a radicand does not represent a negative number raised to an even power. We will make this assumption. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify by factoring:

Multiplying and Simplifying
We have used the product rule for radicals to find products and also to simplify radical expressions. For some radical expressions, it is possible to do both: First find a product and then simplify. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Multiply and simplify.

Dividing Radical Expressions
10.4 Dividing Radical Expressions Dividing and Simplifying Rationalizing Denominators or Numerators With One Term Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Dividing and Simplifying
Just as the root of a product can be expressed as the product of two roots, the root of a quotient can be expressed as the quotient of two roots. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Quotient Rule for Radicals
For any real numbers Remember that an nth root is simplified when its radicand has no factors that are perfect nth powers. Recall too that we assume that no radicands represent negative quantities raised to an even power. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify by taking roots of the numerator and denominator.

Example Divide and, if possible, simplify.

Rationalizing Denominators or Numerators With One Term
When a radical expression appears in a denominator, it can be useful to find an equivalent expression in which the denominator no longer contains a radical. The procedure for finding such an expression is called rationalizing the denominator. We carry this out by multiplying by 1 in either of two ways. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Rationalize each denominator.

Another way to rationalize a denominator is to multiply by 1 outside the radical.

Example Rationalize each denominator.

Sometimes in calculus it is necessary to rationalize a numerator
Sometimes in calculus it is necessary to rationalize a numerator. To do so, we multiply by 1 to make the radicand in the numerator a perfect power. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Expressions Containing Several Radical Terms
10.5 Expressions Containing Several Radical Terms Adding and Subtracting Radical Expressions Products and Quotients of Two or More Radical Terms Rationalizing Denominators or Numerators With Two Terms Terms with Differing Indices Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Adding and Subtracting Radical Expressions
When two radical expressions have the same indices and radicands, they are said to be like radicals. Like radicals can be combined (added or subtracted) in much the same way that we combined like terms earlier in this text. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify by combining like radical terms.

Products and Quotients of Two or More Radical Terms
Radical expressions often contain factors that have more than one term. Multiplying such expressions is similar to finding products of polynomials. Some products will yield like radical terms, which we can now combine. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Multiply. Simplify if possible.

In part (c) of the last example, notice that the inner and outer products in FOIL are opposites, the result, m – n, is not itself a radical expression. Pairs of radical terms like, are called conjugates. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rationalizing Denominators or Numerators With Two Terms
The use of conjugates allows us to rationalize denominators or numerators with two terms. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Rationalize the denominator:

To rationalize a numerator with more than one term, we use the conjugate of the numerator.

Terms with Differing Indices
To multiply or divide radical terms with different indices, we can convert to exponential notation, use the rules for exponents, and then convert back to radical notation. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Simplify Products or Quotients with Differing Indices
1. Convert all radical expressions to exponential notation. 2. When the bases are identical, subtract exponents to divide and add exponents to multiply. This may require finding a common denominator. 3. Convert back to radical notation and, if possible, simplify. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Multiply and, if possible, simplify: Solution
Converting to exponential notation Adding exponents Converting to radical notation Simplifying Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solving Radical Equations
10.6 Solving Radical Equations The Principle of Powers Equations with Two Radical Terms Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Principle of Powers
A radical equation is an equation in which the variable appears in a radicand. Examples are Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Principle of Powers
If a = b, then an = bn for any exponent n. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solve: Solution

Caution! Raising both sides of an equation to an even power may not produce an equivalent equation. In this case, a check is essential. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Solve an Equation with a Radical Term
1. Isolate the radical term on one side of the equation. 2. Use the principle of powers and solve the resulting equation. 3. Check any possible solution in the original equation. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Equations with Two Radical Terms
A strategy for solving equations with two or more radical terms is as follows. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Solve an Equation with Two or More Radical Terms
1. Isolate one of the radical terms. 2. Use the principle of powers. 3. If a radical remains, perform steps (1) and (2) again. 4. Solve the resulting equation. 5. Check the possible solutions in the original equation. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Distance and Midpoint Formulas and Other Applications
10.7 The Distance and Midpoint Formulas and Other Applications Using the Pythagorean Theorem Two Special Triangles The Distance and Midpoint Formulas Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Using the Pythagorean Theorem
There are many kinds of problems that involve powers and roots. Many also involve right triangles and the Pythagorean theorem. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Pythagorean Theorem
In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a2 + b2 = c2. a Leg Hypotenuse c b Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Two Special Triangles When both legs of a right triangle are the same size, we call the triangle an isosceles right triangle. If one leg of an isosceles right triangle has length a then 45° c a a Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Lengths Within Isosceles Right Triangles
The length of the hypotenuse in an isosceles right triangle is the length of a leg times 45o a 45o a Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example The hypotenuse of an isosceles right triangle is 8 ft long. Find the length of a leg. Give an exact answer and an approximation to three decimal places. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Lengths Within 30o – 60o – 90o Right Triangles
The length of the longer leg in a 30o-60o-90o right triangle is the length of the shorter leg times The hypotenuse is twice as long as the shorter leg. 30o 2a 60o a Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example The shorter leg of a 30o-60o-90o right triangle measures 12 in. Find the lengths of the other sides. Give exact answers and, where appropriate, an approximation to three decimal places. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Distance Formula The distance between the points (x1, y1) and (x2, y1) on a horizontal line is |x2 – x1|. Similarly, the distance between the points (x2, y1) and (x2, y2) on a vertical line is |y2 – y1|. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Now consider any two points (x1, y1) and (x2, y2)
Now consider any two points (x1, y1) and (x2, y2) these points, along with (x2, y1), describe a right triangle. The lengths of the legs are |x2 – x1| and |y2 – y1|. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

We find d, the length of the hypotenuse, by using the Pythagorean theorem:
d 2 = |x2 – x1|2 + |y2 – y1|2. Since the square of a number is the same as the square of its opposite, we can replace the absolute-value signs with parentheses: d 2 = (x2 – x1)2 + (y2 – y1)2. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Midpoint Formula The distance formula is needed to verify a formula for the coordinates of the midpoint of a segment connecting two points. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Midpoint Formula If the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are (To locate the midpoint, average the x-coordinates and average the y-coordinates.) y (x2, y2) (x1, y1) x Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

10.8 The Complex Numbers Imaginary and Complex Numbers
Addition and Subtraction Multiplication Conjugates and Division Powers of i Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Imaginary and Complex Numbers
Negative numbers do not have square roots in the real-number system. A larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system. The complex-number system makes use of i, a number that is, by definition, a square root of –1. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Number i i is the unique number for which and i 2 = –1.
We can now define the square root of a negative number as follows: for any positive number p. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Express in terms of i:

Imaginary Numbers An imaginary number is a number that can be written in the form a + bi, where a and b are real numbers and Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The union of the set of all imaginary numbers and the set of all real numbers is the set of all complex numbers. Complex Numbers A complex number is any number that can be written in the form a + bi, where a and b are real numbers. (Note that a and b both can be 0.) Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The following are examples of imaginary numbers:

Addition and Subtraction
The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Add or subtract and simplify.

Conjugate of a Complex Number
The conjugate of a complex number a + bi is a – bi, and the conjugate of a – bi is a + bi. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find the conjugate of each number.

Conjugates and Division
Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Divide and simplify to the form a + bi.

Powers of i Simplifying powers of i can be done by using the fact that i 2 = –1 and expressing the given power of i in terms of i 2. Consider the following: i 23 = (i 2)11i1 = (–1)11i = –i Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify: Solution

Exponents and Radicals

Similar presentations

Presentation on theme: "Exponents and Radicals"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Exponents and Radicals

Similar presentations

Presentation on theme: "Exponents and Radicals"— Presentation transcript:

Similar presentations

About project

Feedback