Geometry Summer Math Packet Review and Study Guide

1 Geometry Summer math PacketReview and Study GuideThis Study Guide is designed to aid students working on the Geometry Summer math Packet . The purpose is to allow students an opportunity to Review some of the ten specific concepts covered in the Summer math Packet . The mastery of these concepts prior to beginning Geometry is essential. For each concept, there are explanations and examples as well as extra problems that students may choose to do on their own if they are experiencing difficulty and would like to reassure themselves that they have indeed mastered the concepts. Students are not required to print out this Guide and may use it simply as an online reference as they complete their Summer AGNES ACADEMYSAINT DOMINIC SCHOOLVERITASA.

2 Order of Operations The rules for Order of Operations are as follows: FIRST: Perform operations inside grouping symbols. Grouping symbols include parentheses ( ), brackets [ ] , braces { } , radical symbols , absolute value symbols and fraction bars. If an expression contains more than one set of grouping symbols, simplify the expression inside the innermost set first. Follow the order of operations within that set of grouping symbols and then work outward. SECOND: Simplify exponents. THIRD: Perform multiplication and division from left to right. (Remember that a fraction bar also indicates division.) FOURTH: Perform addition and subtraction from left to right. Hint: You can use the well-known phrase "Please Excuse My Dear Aunt Sally" to help you remember the Order of Operations.

3 (Remember, however, that multiplication and division must be done in the order that they appear if they do not appear in parentheses. This is also true for addition and subtraction.) Please Parentheses Excuse Exponents My Multiplication Dear Division Aunt Addition Sally Subtraction Example 1 Simplify. 2 4+24 32 y (Note that there are no grouping symbols. Therefore the exponent only applies to the "4" and not the "-".) 16 24 3 216 8 216 160 y Example 2 Simplify. >@22425 5 2425 3425 941664 Example 3 Simplify. 222 253 222 253 (Note that the fraction bar acts as a grouping symbol. You must simplify the numerator and denominator before dividing.) 22 4226213 Try These: Simplify.

4 2. 20 2 4 1 y 1. 1832y 3. 352 823 4. 3501 6. 224 2249 5. 247 3 y Try These Answers: 1. 48 2. 2 3. 1 4. 21 5. -3 6. 2 B. Fractions Adding & Subtracting Fractions To add or subtract fractions, you must always have a common denominator. Once the denominators are the same, you add or subtract only the numerator to get your final answer. The common denominator of choice is the Lowest Common Denominator. If you make an effort to keep the numbers in fractions as small as possible, it will make subsequent calculations much easier. Example 1 Perform the indicated operation. 38416 (Note that in this case it is going to be much easier to have a common denominator of "4" rather than "32", "48" or "64".)

5 324414 Example 2 Perform the indicated operation. 536820 924 242924 Try These: Perform the indicated operation. 1. 8495 2. 3146 3. 7310 8 4. 31 743 12 5. 13 12510 6. 11 75615 Try These Answers: 1. 445 2. 1112 3. 1340 4. 1 5. 65 6. 110 Multiplying Fractions Unlike adding and subtracting, you do not need a common denominator to multiply fractions. To multiply fractions you multiply the numerators and then the denominators. Then, it is good practice to always reduce your final answer. When multiplying a fraction by a whole number, you can rewrite the whole number in fraction form by putting a "1" in the denominator. Example 3 Perform the indicated operation.

6 32453245620310uu u Example 4 Perform the indicated operation. 348431812832u u Try These: Perform the indicated operation. 1. 1427u 2. 7312 4u 3. 12535 4. 32247 5. 42433 u 6. 384u Try These Answers: 1. 27 2. 716 3. 113 4. 37 5. 4 6. 6 Dividing Fractions You have heard this a thousand times: "Dividing by a fraction is the same as multiplying by its reciprocal." To get the reciprocal of a fraction, you switch the numerator and the denominator. Another way of thinking of it is: a fraction multiplied by its reciprocal will always give you "1". For example the reciprocal of 35 is53, and 35153u . As always, to maintain good form, you must reduce your final answer to its simplest form.

7 Example 1 Perform the indicated operation. 3243334298y u Example 2 Perform the indicated operation. 263213619y u Try These: Perform the indicated operation. 1. 11 118 3y 2. 2697y3. 9610 7y 4. 33 475 7 yy 5. 428535uy 6. 11 1618 3 y Try These Answers: 1. 116 2. 727 3. 2120 4. 54 5. 13 6. 11 C. Exponents The properties of exponents are as follows: Product of Powers mn mnxxx 34 722 2 128 Power-of-a-Power: nmmnxx 32677117,649 Power-of-a-Product nnnxyxy 2222339xxx Quotient-of-Powers: mmnnxxx 8853533323 7 Positive Power of a Quotient: nnnaabb 333332446 74 Negative Power of a Quotient: nnnnabba ba 4444233322 8116 Zero Power: 01a 037214xymn Example 1 Simplify each expression leaving no negative exponents.

8 525210103636362xxxxxx Example 2 Simplify each expression leaving no negative exponents. 2332 7621xxyyxyx y Example 3 Simplify each expression leaving no negative exponents. 323236641414164xxxx Example 4 Simplify each expression leaving no negative exponents. 1243333243272645564556455623xyyxy yxyyyxyxyyxy Try These: Simplify each expression leaving no negative exponents. 1. 234stst 2. 4132mn 3. 12324cd 4. 43tv 5. 246476st 6. 432ab Try These Answers: 2. 448116mn 3. 2323cd 4. 4481vt 5. 4444164ts 6. 441681ba D. Radicals Some properties of radicals to know are: Product Property abab 32 16 2 4 2 Quotient Property aabb 555939 A few other things to remember about radicals: #1 You should know what numbers have nice square roots: 4, 9, 16, 25, 36, 49, 64.

9 #2 Never leave a perfect square inside a radical. Example: 32 4 8 2 8 The square root of 8 still has a factor of 4 in it. This answer is incomplete and needs to be finished. #3 Never leave a radical in the denominator of a fraction. Example: 33515555 5 To get rid of the radical in the denominator, multiply the numerator and denominator by the radical Example 1 Evaluate the expression. Leave answer in exact form. 279333 Example 2 Evaluate the expression. Leave answer in exact form. 162641626481 28928 Example 3 Evaluate the expression. Leave answer in exact form. 45 1895 9235 32910 Example 4 Evaluate the expression. Leave answer in exact form.

10 45845 45 9 588435 35 222 22 2310 31022 4 2 Try These: Simplify Evaluate the expression. Leave answer in exact form. 1. 18 2. 89 4. 722825 3. 98 8 5. 208 6. 3655 Try These Answers: 1. 32 2. 223 3. 28 4. 485 5. 102 6. 65 E. Simplifying Expressions To simplify algebraic expressions you need to apply the rules for the Order of Operations and collect like terms. Like terms are terms that contain the same variables raised to the same powers. Constants (numbers with no variable) are also like terms and can be simplified according to the order of operations. Typically, simplified expressions are written with the variables in descending order according to their exponents.