CliffsNotes Algebra II Practice Pack (Cliffnotes) - Softcover

Sterling, Mary Jane

Softcover

ISBN 10: 0470495979 ISBN 13: 9780470495971

Verlag: Cliffs Notes, 2010

Inhaltsangabe

Your guide to a higher score in Algebra II

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Helps you pinpoint where you need the most help and directs you to the corresponding sections of the book

Topic Area Reviews

• Math basics

• Factoring and solving equations

• Function operations and transformations

• Polynomials

• Exponential and logarithmic functions

• Graphing

• Other equations

• Conic sections

• Systems of equations and inequalities

• Systems of linear equations with three or more variables

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Über die Autorin bzw. den Autor

Mary Jane Sterling is a professor of mathematics at Bradley University in Peoria, Illinois. She has been teaching mathematics for over thirty years.

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CliffsNotes Algebra II Practice Pack

By Mary Jane Sterling

John Wiley & Sons

Copyright © 2010 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-49597-1

Chapter One

The Basics

Algebra is a language. You need to know the rules and definitions to understand this language and its many manipulations. In this chapter is a review of some of the important basics of algebra: rules for exponents and operations involving polynomials. These should be reviewed before going on to some of the advanced topics in Algebra II.

Rules for Exponents

A power or exponent tells how many times a number multiplies itself. Many opportunities exist in algebra for combining and simplifying expressions with two or more of these exponential terms in them. The rules used here to combine numbers and variables work for any expression with exponents. They are found in formulas and applications in science, business, and technology, as well as math. The term [a.sup.4] has an exponent of 4 and a base of a. The base is what gets multiplied repeatedly. The exponent tells how many times that repeatedly is.

Laws for Using Exponents

[a.sup.n] x [a.sup.m] = [an.sup.+m] When multiplying two numbers that have the same base, add their exponents.

[a.sup.n]/[a.sup.m] = [a.sup.n-m] When dividing two numbers that have the same base, subtract their exponents.

[([a.sup.n]).sup.m] = [a.sup.n m] When raising a value that has an exponent to another power, multiply the two exponents.

[(a b).sup.n] = [a.sup.n] [b.sup.n] The product of two numbers raised to a power is equal to raising each number to that power and then multiplying them together.

[(a/b).sup.n] = [a.sup.n]/[b.sup.n] The quotient of two numbers raised to a power is equal to raising each of the numbers to that power and then dividing them.

[a.sup.-n] = 1/[a.sup.n] A value raised to a negative power can be written as a fraction with the positive power of that number in the denominator.

[a.sup.0] = 1 Any number (except 0) raised to the 0 power is equal to 1.

Example Problems

These problems show the answers and solutions.

1. Simplify: [x.sup.4] ([x.sup.3/[x.sup.-2]).sup.3]

answer: [x.sup.19]

In this case, the course of action is to simplify the expressions inside the parentheses first, raise that result to the third power, and finally multiply by the first factor.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. Simplify: [y.sup.-3] [([y.sup.2]).sup.4] [y.sup.2]/[yy.sup.3]

answer: [y.sup.5]

The denominator reads [yy.sup.3], which implies that the first factor has an exponent of 1, reading [y.sup.1] [y.sup.3]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. Simplify: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

answer: [a.sup.23]/[b.sup.6]

A nice property of fractions is that when they're raised to a negative power, you can rewrite the expression and change the power to a positive if you "flip" the fraction. So [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. First, rewrite the second fraction without the negative exponent. Then simplify the fractions inside the parentheses. The next step is to raise the factors in the parentheses to the powers. Lastly, multiply the terms in the two numerators and denominators.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Work Problems

Use these problems to give yourself additional practice.

1. Simplify: [([x.sup.3]/[x.sup.-3]).sup.2]

2. Simplify: [([y.sup.4]/3[x.sup.2]).sup.-3]

3. Simplify: [a.sup.-2] [x.sup.2]/[a.sup.4] x [([a.sup.3]/[x.sup.3]).sup.4]

4. Simplify: [(ab[c.sup.2]).sup.4]/[a.sup.2]b[c.sup.-1]

5. Simplify: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Worked Solutions

1. [x.sup.12] First simplify inside the parentheses. Then raise the result to the second power.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. 27[x.sup.6]/[y.sup.12] First "flip" the fraction and change the power to positive.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. [a.sup.6]/[x.sup.11] First raise the factors in the parentheses to the fourth power. Then simplify the first fraction before multiplying the two fractions together.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4. [a.sup.2][b.sup.3][c.sup.9] First raise the numerator to the fourth power. Then simplify the fraction.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5. [x.sup.4][z.sup.20]/16 Since both fractions are raised to the fourth power, it is easier to combine them in the same parentheses and then later raise the result to the fourth power.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Adding and Subtracting Polynomials

One major objective of working with algebraic expressions is to write them as simply as possible and in a logical, generally accepted arrangement. When more than one term exists (a term consists of one or more factors multiplied together and separated from other terms by + or -), then you check to see whether they can be combined with other terms that are like them. Numbers by themselves without letters or variables are like terms. You can combine 14 and 8 because you know what they are and know the rules. For instance, 14 + 8 = 22, 14 - 8 = 6, 14(8) = 112, and so on. Numbers can be written so they can combine with one another. They can be added, subtracted, multiplied, and divided, as long as you don't divide by zero. Fractions can be added if they have a common denominator. Algebraic expressions involving variables or letters have to be dealt with carefully. Since the numbers that the letters represent aren't usually known, you can't add or subtract terms with different letters. The expression 2a + 3b has to stay that way. That's as simple as you can write it, but the expression 4c + 3c can be simplified. You don't know what c represents, but you can combine the terms to tell how many of them you have (even though you don't know what they are!): 4c + 3c = 7c. Here are some other examples:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice that there are two different kinds of terms, one with the x squared and the other with the y squared. Only those that have the letters exactly alike with the exact same powers can be combined. The only thing affected by adding and subtracting these terms is the coefficient.

Example Problems

These problems show the answers and solutions.

1. Simplify 5x[y.sup.2] + 8x -...

Details zur Ausgabe

• Verlag: Cliffs Notes
• Erscheinungsdatum: 2010
• Sprache: Englisch
• ISBN 10: 0470495979
• ISBN 13: 9780470495971
• Einband: Taschenbuch
• Anzahl der Seiten: 264
• Kontakt zum Hersteller: HOUGHTON MIFFLIN; https://www.hmhco.com/shop; ; 125 HIGH STREET; BOSTON; MA; 2110; USA
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