Algebra I Workbook For Dummies Read online

  1. Simplify .

  Solve It

  2. Simplify .

  Solve It

  3. Simplify .

  Solve It

  4. Simplify .

  Solve It

  5. Simplify the radicals in before adding.

  Solve It

  6. Simplify before subtracting.

  Solve It

  Rationalizing Fractions

  You rationalize a fraction with a radical in its denominator (bottom) by changing the original fraction to an equivalent fraction that has a multiple of that radical in the numerator (top). Usually, you want to remove radicals from the denominator. The square root of a number that isn’t a perfect square is irrational. Dividing with an irrational number is difficult because, when expressed as decimals, those numbers never end and never have a repeating pattern.

  To rationalize a fraction with a square root in the denominator, multiply both the numerator and denominator by that square root.

  Q.

  A. . Recall the property that . It works both ways: .

  Multiplying the denominator by itself creates a perfect square (so there’ll be no radical). Simplify and reduce the fraction.

  Q.

  A. . In this problem, you multiply both of the radicals by the radical in the denominator. The products lead to results that can be simplified nicely.

  7. Rationalize .

  Solve It

  8. Rationalize .

  Solve It

  9. Rationalize .

  Solve It

  10. Rationalize .

  Solve It

  Arranging Radicals as Exponential Terms

  The radical symbol indicates that you’re to do the operation of taking a root — or figuring out what number multiplied itself to give you the value under the radical. An alternate notation, a fractional exponent, also indicates that you’re to take a root, but fractional exponents are much more efficient when you perform operations involving powers of the same number.

  The equivalence between the square root of a and the fractional power notation is . The 2 in the bottom of the fractional exponent indicates a square root. The general equivalence between all roots, powers, and fractional exponents is .

  Q.

  A. . The root is 3; you’re taking a cube root. The 3 goes in the fraction’s denominator. The 7 goes in the fraction’s numerator.

  Q.

  A. . The exponent becomes negative when you bring up the factor from the fraction’s denominator. Also, when no root is showing on the radical, it’s assumed that a 2 goes there because it’s a square root.

  11. Write the radical form in exponential form:

  Solve It

  12. Write the radical form in exponential form:

  Solve It

  13. Write the radical form in exponential form:

  Solve It

  14. Write the radical form in exponential form (assume that y is positive):

  Solve It

  15. Write the radical form in exponential form (assume that x is positive):

  Solve It

  16. Write the radical form in exponential form:

  Solve It

  Using Fractional Exponents

  Fractional exponents by themselves are fine and dandy. They’re a nice, compact way of writing an operation to be performed on the power of a number. What’s even nicer is when you can simplify or evaluate an expression, and its result is an integer. You want to take advantage of these simplification situations.

  If a value is written am/n, the easiest way to evaluate it is to take the root first and then raise the result to the power. Doing so keeps the numbers relatively small — or at least smaller than the power might become. The answer comes out the same either way. Being able to compute these in your head saves time.

  Q. 84/3 =

  A. 16. Finding the cube root first is easier than raising 8 to the fourth power, which is 4,096, and then taking the cube root of that big number. By finding the cube root first, you can do all the math in your head. If you write out the solution, here’s what it would look like:

  Q.

  A. . Use the rule from Chapter 4 on raising a fraction to a power. When the number 1 is raised to any power, the result is always 1. The rest involves the denominator.

  17. Compute the value of 45/2.

  Solve It

  18. Compute the value of 272/3.

  Solve It

  19. Compute the value of .

  Solve It

  20. Compute the value of .

  Solve It

  Simplifying Expressions with Exponents

  Writing expressions using fractional exponents is better than writing them as radicals because fractional exponents are easier to work with in situations where something complicated or messy needs to be simplified into something neater. The simplifying is done when you multiply and/or divide factors with the same base. When the bases are the same, you use the rules for multiplying (add exponents), dividing (subtract exponents), and raising to powers (multiply exponents). Refer to Chapter 4 if you need a reminder on these concepts. Here are some examples:

  Q. 24/3 ´ 25/3 =

  A. 8. Remember, when numbers with the same base are multiplied together, you add the exponents.

  24/3 × 25/3 = 24/3 + 5/3 = 29/3 = 23 = 8

  Q.

  A. 625. Notice that the numbers don’t have the same base! But 25 is a power of 5, so you can rewrite it and then apply the fourth root.

  21. Simplify 21/4 × 23/4.

  Solve It

  22. Simplify .

  Solve It

  23. Simplify .

  Solve It

  24. Simplify .

  Estimating Answers

  Radicals appear in many mathematical applications. You need to simplify radical expressions, but it’s also important to have an approximate answer in mind before you start. Doing so lets you evaluate whether the answer makes sense, based on your estimate. If you just keep in mind that is about 1.4, is about 1.7 and is about 2.2, you can estimate many radical values. Here are some examples:

  Q. Estimate the value of .

  A. About 14. Simplifying the radical, you get . If is about 1.4, then 10(1.4) is 14.

  Q. Estimate the value of .

  A. About 9.5. Simplifying the radicals, you get . If is about 2.2, then 2(2.2) is 4.4. Multi-plying 3(1.7) for 3 times root three, you get 5.1. The sum of 4.4 and 5.1 is 9.5.

  25. Estimate .

  Solve It

  26. Estimate .

  Solve It

  27. Estimate .

  Solve It

  28. Estimate .

  Solve It

  Answers to Problems on Radicals

  This section provides the answers (in bold) to the practice problems in this chapter.

  1. Simplify . The answer is .

  2. Simplify . The answer is .

  3. Simplify . The answer is .

  4. Simplify . The answer is .

  5. Simplify the radicals in before adding. The answer is .

  6. Simplify before subtracting. The answer is .

  7. Rationalize . The answer is .

  8. Rationalize . The answer is .

  9. Rationalize . The answer is .

  10. Rationalize . The answer is .

  11. Write the radical form in exponential form: . The answer is .

  12. Write the radical form in exponential form: . The answer is .

  13. Write the radical form in exponential form: . The answer is .

  14. Write the radical form in exponential form: . The answer is .

  15. Write the radical form in exponential form: . The answer is .

  16. Write the radical form in exponential form: . The answer is .

  17. Compute the value of . The answer is 32.

  18. Compute the value of .The answer is 9.

  19. Compute the value of . The answer is .

  20. Compute the value of . The answer is .

  21. Simplify . The answer is 2.

  22. S
implify . The answer is 36.

  23. Simplify . The answer is 8.

  24. Simplify . The answer is .

  25. Estimate . The answer is about 5.6.

  26. Estimate . The answer is about 11.

  27. Estimate . The answer is about 7.6.

  28. Estimate . The answer is about 12.32.

  Chapter 6

  Simplifying Algebraic Expressions

  In This Chapter

  Making algebraic expressions more user-friendly

  Maintaining order with the order of operations

  Combining all the rules for simpler computing

  The operations of addition, subtraction, multiplication, and division are familiar to every grade school student. Fortunately, these operations work the same, no matter what level or what kind of math you do. As I explain in this chapter, though, algebra and its properties introduce some new twists to those elementary rules. The good news is that you can apply all these basic rules to the letters, which stand for variables, in algebra. However, because you usually don’t know what values the variables represent, you have to be careful when doing the operations and reporting the results. This chapter offers you multitudes of problems to make sure you keep everything in order.

  The most commonly used variable in algebra is x. Because the variable x looks so much like the times sign, ×, other multiplication symbols are used in algebra problems. The following are all equivalent multiplications:

  In spreadsheets and calculators, the * sign indicates multiplication.

  Adding and Subtracting Like Terms

  In algebra, the expression like terms refers to a common structure for the terms under consideration. Like terms have exactly the same variables in them, and each variable is “powered” the same (if x is squared and y cubed in one term, then x squared and y cubed occur in a like term). When adding and subtracting algebraic terms, the terms must be alike, with the same variables raised to exactly the same power, but the numerical coefficients can be different. For example, two terms that are alike are 2a3b and 5a3b. Two terms that aren’t alike are 3xyz and 4x2yz, where the power on the x term is different in the two terms.

  Q. 6a + 2b – 4ab + 7b + 5ab – a + 7 =

  A. 5a + 9b + ab + 7. First, change the order and group the like terms together; then compute:

  6a + 2b – 4ab + 7b + 5ab – a + 7 =(6a – a) + (2b + 7b) + (–4ab + 5ab) + 7

  The parentheses aren’t necessary, but they help to keep track of what you can combine.

  Q. 8x2 – 3x + 4xy – 9x2 – 5x – 20xy =

  A. –x2 – 8x – 16xy. Again, combine like terms and compute:

  8x2 – 3x + 4xy – 9x2 – 5x – 20xy = (8x2 – 9x2) + (–3x – 5x) + (4xy – 20xy) = –x2 – 8x – 16xy

  1. Combine the like terms in 4a + 3ab – 2ab + 6a.

  Solve It

  2. Combine the like terms in 3x2y – 2xy2 + 4x3 – 8x2y.

  Solve It

  3. Combine the like terms in 2a2 + 3a – 4 + 7a2 – 6a + 5.

  Solve It

  4. Combine the like terms in ab +bc + cd + de – ab + 2bc + e.

  Solve It

  Multiplying and Dividing Algebraically

  Multiplying and dividing algebraic expressions is somewhat different from adding and subtracting them. When multiplying and dividing, the terms don’t have to be exactly alike. You can multiply or divide all variables with the same base — using the laws of exponents (check out Chapter 4 for more information) — and you multiply or divide the number factors.

  If a variable’s power is greater in the denominator, then the difference between the two powers is preferably written as a positive power of the base — in the denominator — instead of with a negative exponent in the numerator. For example:

  Q. (4x2y2z3)(3xy4z3) =

  A. 12x3y6z6. The product of 4 and 3 is 12. Multiply the x’s to get x2(x) = x3. Multiply the y’s and then the z’s and you get y2(y4) = y6 and z3(z3) = z6. Each variable has its own power determined by the factors multiplied together to get it.

  Q.

  A. . The power of x in the numerator is greater than that in the denominator, and y has the greater power in the denominator. The only factor of z is in the numerator, so it stays there.

  5. Multiply (3x)(2x2).

  Solve It

  6. Multiply (4y2)( –x4y).

  Solve It

  7. Multiply (6x3y2z2)(8x3y4z).

  Solve It

  8. Divide (write all exponents as positive numbers) .

  Solve It

  9. Divide (write all exponents as positive numbers) .

  Solve It

  10. Divide (write all exponents as positive numbers) .

  Solve It

  Incorporating Order of Operations

  Because much of algebra involves symbols for numbers, operations, relationships, and groupings, are you really surprised that order is something special, too? Order helps you solve problems by making sure that everyone doing the problem follows the same procedure and gets the same answer. Order is how mathematicians have been able to communicate — agreeing on these same conventions. The order of operations is just one of many such agreements that have been adopted.

  When you perform operations on algebraic expressions and you have a choice between one or more operations to perform, use the following order:

  1. Perform all powers and roots, moving left to right.

  2. Perform all multiplication and division, moving left to right.

  3. Perform all addition and subtraction, moving left to right.

  These rules are interrupted if the problem has grouping symbols. You first need to perform operations in grouping symbols, such as ( ), { }, [ ] , above and below fraction lines, and inside radicals.

  Q. 2 × 4 – 10 ÷ 5 =

  A. 6. First do the multiplication and division and then subtract the results:

  2 × 4 – 10 ÷ 5 = 8 – 2 = 6

  Q.

  A. 4. First find the values of the power and root (22 = 4 and ). Then multiply in the numerator. After you add the two terms in the numerator and subtract in the denominator, then you can perform the final division. Here’s how it breaks down:

  11.

  Solve It

  12.

  Solve It

  13. 2 × 33 + 3(22 – 5) =

  Solve It

  14.

  Solve It

  Evaluating Expressions

  Evaluating an expression means that you want to change it from a bunch of letters and numbers to a specific value — some number. After you solve an equation or inequality, you want to go back and check to see whether your solution really works — so you evaluate the expression. For example, if you let x = 2 in the expression 3x2 – 2x + 1, you replace all the x’s with 2s and apply the order of operations when doing the calculations. In this case, you get 3(2)2 – 2(2) + 1 = 3(4) – 4 + 1 = 12 – 4 + 1 = 9. Can you see why knowing that you square the 2 before multiplying by the 3 is so important? If you multiply by the 3 first, you end up with that first term being worth 36 instead of 12. It makes a big difference.