The sat skill exam 5 pdf

Sometimes, you will see an exponent with a vari- able: b n . The “b”represents a number that will be a fac- tor to itself “n” times. Example: b n where b = 5 and n = 3 Don’t let the variables fool you. Most expressions are very easy once you substi- tute in numbers. b n = 5 3 = 5 × 5 × 5 = 125 Laws of Exponents ■ Any base to the zero power is always 1. Examples: 5 0 = 1 70 0 = 1 29,874 0 = 1 ■ When multiplying identical bases, you add the exponents. Examples: 2 2 × 2 4 × 2 6 = 2 12 a 2 × a 3 × a 5 = a 10 ■ When dividing identical bases, you subtract the exponents. Examples: ᎏ 2 2 5 3 ᎏ = 2 2 ᎏ a a 7 4 ᎏ = a 3 Here is another method of illustrating multipli- cation and division of exponents: b m × b n = b m + n ᎏ b b m n ᎏ = b m – n ■ If an exponent appears outside of the parentheses, you multiply the exponents together. Examples: (3 3 ) 7 = 3 21 (g 4 ) 3 = g 12 Squares and Square Roots The square root of a number is the product of a num- ber and itself. For example, in the expression 3 2 = 3 × 3 = 9, the number 9 is the square of the number 3. If we reverse the process, we can say that the number 3 is the square root of the number 9. The symbol for square root is ͙25 ෆ and it is called the radical. The number inside of the radical is called the radicand. Example: 5 2 = 25; therefore, ͙25 ෆ = 5 Since 25 is the square of 5, we also know that 5 is the square root of 25. Perfect Squares The square root of a number might not be a whole number. For example, the square root of 7 is 2.645751311 It is not possible to find a whole number that can be multiplied by itself to equal 7. A whole number is a perfect square if its square root is also a whole number. Examples of perfect squares: 1,4,9,16,36,49,64,81,100, Properties of Square Root Radicals ■ The product of the square roots of two numbers is the same as the square root of their product. Example: ͙a ෆ × ͙b ෆ = ͙a × b ෆ ͙5 ෆ × ͙3 ෆ = ͙15ෆ ■ The quotient of the square roots of two numbers is the square root of the quotient. Example: ■ The square of a square root radical is the radicand. Example: (͙N ෆ ) 2 = N (͙3 ෆ ) 2 = ͙3 ෆ × ͙3 ෆ = ͙9 ෆ = 3 √ ¯¯¯ √ ¯¯¯ a √ ¯¯¯ b √ ¯¯ ¯ 5 = = a b (b ≠ 0) √ ¯¯¯¯¯ 15 √ ¯¯¯ 3 √ ¯¯¯¯¯ 15 3 = –THE SAT MATH SECTION– 112 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 112 ■ To combine square root radicals with the same radicands, combine their coefficients and keep the same radical factor. You may add or subtract radicals with the same radicand. Example: a͙b ෆ + c͙b ෆ = (a + c)͙b ෆ 4͙3 ෆ + 2͙3 ෆ = 6͙3 ෆ ■ Radicals cannot be combined using addition and subtraction. Example: ͙a + b ෆ ≠ ͙a ෆ + ͙b ෆ ͙4 + 11 ෆ ≠ ͙4 ෆ + ͙11 ෆ ■ To simplify a square root radical, write the radi- cand as the product of two factors, with one num- ber being the largest perfect square factor. Then write the radical over each factor and simplify. Example: ͙8 ෆ = ͙4 ෆ × ͙2 ෆ = 2͙2 ෆ Integer and Rational Exponents Integer Exponents When dealing with negative exponents, remember that a –n = ᎏ a 1 n ᎏ . Examples: 4 –2 = ᎏ 4 1 2 ᎏ = ᎏ 1 1 6 ᎏ –2 –3 = ᎏ – 1 2 3 ᎏ = ᎏ – 1 8 ᎏ = – ᎏ 1 8 ᎏ Rational Exponents Recall that rational numbers are all numbers that can be written as fractions ( ᎏ 2 3 ᎏ ), terminating decimals (.75), and repeating decimals (.666 . . . ). Keeping this in mind, it’s no surprise that numbers raised to rational exponents are just numbers raised to a fractional power. What is the value of 4 ᎏ 1 2 ᎏ ? 4 ᎏ 1 2 ᎏ can be rewritten as ͙4 ෆ , so it is equal to 2. Any time you see a number with a fractional exponent, the numerator of that exponent is the power you raise the number to, and the denominator is the root you take. Examples: 25 = ͙ 2 25 1 ෆ 8= ͙ 3 8 1 ෆ 16 = ͙ 2 16 1 ෆ Divisibility and Factors Like multiplication, division can be represented in a few different ways: 8 ÷ 3 3ͤ8 ෆ ᎏ 8 3 ᎏ In each of the above, 3 is the divisor and 8 is the dividend. Odd and Even Numbers An even number is a number that can be divided by the number 2:2,4,6,8,10,12,14, An odd number can- not be divided evenly by the number 2: 1, 3, 5, 7, 9, 11, 13, The even and odd numbers listed are also exam- ples of consecutive even numbers and consecutive odd numbers because they differ by two. Here are some helpful rules for how even and odd numbers behave when added or multiplied: even + even = even and even × even = even odd + odd = even and odd × odd = odd odd + even = odd and even × odd = even Dividing by Zero Dividing by zero is not possible. This is important to remember when solving for a variable in the denomi- nator of a fraction. Example: ᎏ a – 6 3 ᎏ In this problem, we know that a cannot be equal to 3, because that would yield a zero in the denominator. a – 3 = 0 a ≠ 3 ᎏ 1 2 ᎏ ᎏ 1 3 ᎏ ᎏ 1 2 ᎏ –THE SAT MATH SECTION– 113 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 113 Factors and Multiples Factors are numbers that can be divided into a larger number without a remainder. Example: 12 ÷ 3 = 4 The number 3 is, therefore, a factor of the number 12. Other factors of 12 are 1, 2, 4, 6, and 12. The common factor of two numbers are the fac- tors that both numbers have in common. Example: The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 18 = 1, 2, 3, 6, 9, and 18. From the above, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. From this list, we can also determine that the greatest common factor of 24 and 18 is 6. Determining the greatest common fac- tor is useful for reducing fractions. Any number that can be obtained by multiplying a number x by a positive integer is called a multiple of x. Example: Some multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40 . . . Some multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56 . . . From the above, you can also determine that the least common multiple of the numbers 5 and 7 is 35. The least common multiple, or LCM, is used when performing various operations with fractions. Prime and Composite Numbers A positive integer that is greater than the number 1 is either prime or composite, but not both. ■ A prime number has only itself and the number 1 as factors. Examples: 2,3,5,7,11,13,17,19,23, ■ A composite number is a number that has more than two factors. Examples: 4,6,8,9,10,12,14,15,16, ■ The number 1 is neither prime nor composite. Prime Factorization The SAT will ask you to combine several skills at once. One example of this, called prime factorization, is a process of breaking down factors into prime numbers. Examples: 18 = 9 × 2 The number 9 can also be written as 3 × 3. So, the prime factoriza- tion of 18 is: 18 = 3 × 3 × 2 This can also be demonstrated with the factors 6 and 3: 18 = 6 × 3 Because we know that 6 is equal to 2 × 3, we can write: 18 = 2 × 3 × 3 According to the commutative law, we know that 3 × 3 × 2 = 2 × 3 × 3. Number Lines and Signed Numbers You have surely dealt with number lines in your career as a math student. The concept of the number line is simple: Less than is to the left and greater than is to the right . . . Sometimes, however, it is easy to get confused about the value of negative numbers. To keep things simple, remember this rule: If a > b, then –b > –a. Example: If 7 > 5, then –5 > –7. –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 Greater Than Less Than –THE SAT MATH SECTION– 114 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 114 Absolute Value The absolute value of a number or expression is always positive because it is the distance a number is away from zero on a number line. Example: ԽϪ1Խϭ1 Խ2 Ϫ 4ԽϭԽϪ2Խϭ2 Working with Integers Multiplying and Dividing Here are some rules for working with integers: (+) × (+) = + (+) Ϭ (+) = + (+) × (–) = – (+) Ϭ (–) = – (–) × (–) = + (–) Ϭ (–) = + A simple rule for remembering the above is that if the signs are the same when multiplying or dividing, the answer will be positive and if the signs are different, the answer will be negative. Adding Adding the same sign results in a sum of the same sign: (+) + (+) = + and (–) + (–) = – When adding numbers of different signs, follow this two-step process: 1. Subtract the absolute values of the numbers. 2. Keep the sign of the larger number. Examples: –2 + 3 = 1. Subtract the absolute values of the numbers: 3 – 2 = 1 2. The sign of the larger number (3) was originally positive, so the answer is positive 1. 8 + –11 = 1. Subtract the absolute values of the numbers: 11 – 8 = 3 2. The sign of the larger number (11) was originally negative, so the answer is –3. Subtracting When subtracting integers, change all subtraction to addition and change the sign of the number being sub- tracted to its opposite. Then follow the rules for addition. Examples: (+10) – (+12) = (+10) + (–12) = –2 (–5) – (–7) = (–5) + (+7) = +2 Decimals The most important thing to remember about decimals is that the first place value to the right begins with tenths. The place values are as follows: In expanded form, this number can also be expressed as . . . 1,268.3457 = (1 × 1,000) + (2 × 100) + (6 × 10) + (8 × 1) + (3 × .1) + (4 × .01) + (5 × .001) + (7 × .0001) 1 T H O U S A N D S 2 H U N D R E D S 6 T E N S 8 O N E S • D E C I M A L 3 T E N T H S 4 H U N D R E D T H S 5 T H O U S A N D T H S 7 T E N T H O U S A N D T H S POINT –THE SAT MATH SECTION– 115 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 115 Comparing Decimals Comparing decimals is actually quite simple. Just line up the decimal points and fill in any zeroes needed to have an equal number of digits. Example: Compare .5 and .005. Line up decimal points .500 and add zeroes. .005 Then ignore the decimal point and ask, which is bigger: 500 or 5? 500 is definitely bigger than 5, so .5 is larger than .005. Fractions To do well when working with fractions, it is necessary to understand some basic concepts. Here are some math rules for fractions using variables: ᎏ a b ᎏ × ᎏ d c ᎏ = ᎏ b a × × d c ᎏ ᎏ a b ᎏ + ᎏ b c ᎏ = ᎏ a + b c ᎏ ᎏ a b ᎏ ÷ ᎏ d c ᎏ = ᎏ a b ᎏ × ᎏ d c ᎏ = ᎏ a b × × d c ᎏ ᎏ a b ᎏ + ᎏ d c ᎏ = ᎏ ad b + d bc ᎏ Multiplying Fractions Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the numerators and the denominators, writing each in the respective place over or under the fraction bar. Example: ᎏ 4 5 ᎏ × ᎏ 6 7 ᎏ = ᎏ 2 3 4 5 ᎏ Dividing Fractions Dividing fractions is the same thing as multiplying fractions by their reciprocal. To find the reciprocal of any number, switch its numerator and denominator. For example, the reciprocals of the following numbers are: ᎏ 1 3 ᎏ = ᎏ 3 1 ᎏ = 3 x = ᎏ 1 x ᎏᎏ 4 5 ᎏ = ᎏ 5 4 ᎏ 5 = ᎏ 1 5 ᎏ When dividing fractions, simply multiply either fraction by the other’s reciprocal to get the answer. Example: ᎏ 1 2 2 1 ᎏ ÷ ᎏ 3 4 ᎏ = ᎏ 1 2 2 1 ᎏ × ᎏ 4 3 ᎏ = ᎏ 4 6 8 3 ᎏ = ᎏ 1 2 6 1 ᎏ Adding and Subtracting Fractions ■ To add or subtract fractions with like denomina- tors, just add or subtract the numerators and leave the denominator as it is. For example, ᎏ 1 7 ᎏ + ᎏ 5 7 ᎏ = ᎏ 6 7 ᎏ and ᎏ 5 8 ᎏ – ᎏ 2 8 ᎏ = ᎏ 3 8 ᎏ ■ To add or subtract fractions with unlike denomi- nators, you must find the least common denominator, or LCD. For example, if given the denominators 8 and 12, 24 would be the LCD because 8 × 3 = 24 and 12 × 2 = 24. In other words, the LCD is the smallest number divis- ible by each of the denominators. Once you know the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the necessary number to get the LCD, and then add or subtract the new numerators. Example: ᎏ 1 3 ᎏ + ᎏ 2 5 ᎏ = ᎏ 5 5 ( ( 1 3 ) ) ᎏ + ᎏ 3 3 ( ( 2 5 ) ) ᎏ = ᎏ 1 5 5 ᎏ + ᎏ 1 6 5 ᎏ = ᎏ 1 1 1 5 ᎏ Sets Sets are collections of numbers and are usually based on certain criteria. All the numbers within a set are called the members of the set. For example, the set of integers looks like this: { –3,–2 ,–1,0,1,2,3, } The set of whole numbers looks like this: { 0,1,2,3, } –THE SAT MATH SECTION– 116 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 116 When you find the elements that two (or more) sets have in common, you are finding the intersection of the sets. The symbol for intersection is: ∩. For example, the intersection of the integers and the whole numbers is the set of the whole numbers itself. This is because the elements (numbers) that they have in common are {0,1,2,3, }.Consider the set of positive even integers and the set of positive odd integers. The positive even integers are: {2,4,6,8, } The positive odd integers are: {1,3,5,7, } If we were to combine the set of positive even numbers with the set of positive odd numbers, we would have the union of the sets: {1,2,3,4,5, } The symbol for union is: ∪. Mean, Median, and Mode To find the average or mean of a set of numbers, add all of the numbers together and divide by the quantity of numbers in the set. Average = ᎏ qu n a u n m ti b ty er o s f e s t et ᎏ Example: Find the average of 9, 4, 7, 6, and 4. ᎏ 9+4+7 5 +6+4 ᎏ = ᎏ 3 5 0 ᎏ = 6 (because there are 5 numbers in the set) To find the median of a set of numbers, arrange the numbers in ascending order and find the middle value. ■ If the set contains an odd number of elements, then simply choose the middle value. Example: Find the median of the number set: 1, 5, 3, 7, 2. First, arrange the set in ascending order: 1, 2, 3, 5, 7, and then, choose the middle value: 3. The answer is 3. ■ If the set contains an even number of elements, simply average the two middle values. Example: Find the median of the number set: 1, 5, 3, 7, 2, 8. First, arrange the set in ascending order: 1, 2, 3, 5, 7, 8, and then, choose the middle values 3 and 5. Find the average of the numbers 3 and 5: ᎏ 3+ 2 5 ᎏ = 4. The answer is 4. The mode of a set of numbers is the number that occurs the greatest number of times. Example: For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the number 3 is the mode because it occurs the most number of times. Percent A percent is a measure of a part to a whole, with the whole being equal to 100. ■ To change a decimal to a percentage, move the decimal point two units to the right and add a percentage symbol. Examples: .45 = 45% .07 = 7% .9 = 90% ■ To change a percentage to a decimal, simply move the decimal point two places to the left and elimi- nate the percentage symbol. Examples: 64% = .64 87% = .87 7% = .07 ■ To change a fraction to a percentage, first change the fraction to a decimal. To do this, divide the numerator by the denominator. Then change the decimal to a percentage. Examples: ᎏ 4 5 ᎏ = .80 = 80% ᎏ 2 5 ᎏ = .4 = 40% ᎏ 1 8 ᎏ = .125 = 12.5% –THE SAT MATH SECTION– 117 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 117 . is ͙ 25 ෆ and it is called the radical. The number inside of the radical is called the radicand. Example: 5 2 = 25; therefore, ͙ 25 ෆ = 5 Since 25 is the square of 5, we also know that 5 is the. the denominator. Then change the decimal to a percentage. Examples: ᎏ 4 5 ᎏ = .80 = 80% ᎏ 2 5 ᎏ = .4 = 40% ᎏ 1 8 ᎏ = .1 25 = 12 .5% THE SAT MATH SECTION– 117 56 58 SAT2 006[04](fin).qx 11/21/ 05. digits. Example: Compare .5 and .0 05. Line up decimal points .50 0 and add zeroes. .0 05 Then ignore the decimal point and ask, which is bigger: 50 0 or 5? 50 0 is definitely bigger than 5, so .5 is