The document discusses different types of numbers and operations involving positive and negative numbers. It explains rules for addition, subtraction, multiplication, and division of positive and negative numbers. It also covers order of operations using PEMDAS and provides examples of solving expressions using proper order. Finally, it discusses properties and rules for exponents, including adding, subtracting, multiplying, and dividing terms with the same base and combining exponents.

1. MATH COMPUTATION SKILLS Section I DePaul Math Placement Test

4. The most basic form of mathematical expressions involving several mathematical operations can only be solved by using the order of PEMDAS. This catchy acronym stands for: P arentheses: first, perform the operations in the innermost parentheses. A set of parentheses supercedes any other operation. E xponents: before you do any other operation, raise all the required bases to the prescribed exponent. Exponents include roots, since root operations are the equivalent of raising a base to the 1 / n, where ‘n’ is any integer. M ultiplication and D ivision: perform multiplication and division. A ddition and S ubtraction: perform addition and subtraction. Order of Operations

5. Examples Order of Operations Here are two examples illustrating the usage of PEMDAS. Let’s work through a few examples to see how order of operations and PEMDAS work. 3 X 2 3 + 6 4. Since nothing is enclosed in parentheses, the first operation we carry out is exponentiation: 3 X2 3 + 6÷4= 38+6÷4 Next, we do all the necessary multiplication and division: 38+6÷4= 24÷1.5 Lastly, we perform the required addition and subtraction. Our final answer is: 24÷1.5= 25.5 Here a few question to try yourself. Evaluate: 6 (2 3 2(5-3)) Hint: Start solving from the innermost parenthesis first (Final Answer is 12). Here’s another question. Evaluate: 5-2 2 ⁄6+4. Hint: Solve the numerator and denominator separately. Order of Operations

6. An exponent defines the number of times a number is to be multiplied by itself. For example, in a b , where a is the base and b the exponent, a is multiplied by itself b times. In a numerical example, 2 5 = 2 x 2 x 2 x 2 x 2. An exponent can also be referred to as a power. The following are other terms related to exponents with which you should be familiar: Base. The base refers to the 3 in 3 5 . It is the number that is being multiplied by itself however many times specified by the exponent. Exponent. The exponent is the 5 in 3 5 . It indicates the number of times the base is to be multiplied with itself. Square. Saying that a number is squared means that the number has been raised to the second power, i.e., that it has an exponent of 2. In the expression 6 2 , 6 has been squared. Cube. Saying that a number is cubed means that it has been raised to the third power, i.e., that it has an exponent of 3. In the expression 4 3 , 4 has been cubed. Exponents

7. In order to add or subtract numbers with exponents, you must first find the value of each power, then add the two numbers. For example, to add 3 3 + 4 2 , you must expand the exponents to get (3x 3 x 3) + (4 x 4), and then, 27 + 16 = 43. However, algebraic expressions that have the same bases and exponents, such as 3 x 4 and 5 x 4 , can be added and subtracted. For example, 3 x 4 + 5 x 4 = 8 x 4 . Adding and Subtracting Numbers with Exponents

8. To multiply exponential numbers raised to the same exponent, raise their product to that exponent: a n x b n = (ax b) n = (ab) n 4 3 x 5 3 =(4x5) 3 = 20 3 To divide exponential numbers raised to the same exponent, raise their quotient to that exponent: a n /b n = (a/b) n 12 5 / 3 5 = (12/3) 5 = 4 3 To multiply exponential numbers or tems that have the same base, add the exponents together: a m b n = (ab) (m+n) 3 6 x3 2 = 3 (6+2) = 3 8 To divide two same-base exponential numbers or terms, just subtract the exponents: a m /b n = (a/b) (m-n) 3 6 /3 2 = 3 (6-2) = 3 4 When an exponent is raised to another exponent In cases, (3 2 ) 4 and ( x 4 ) 3 . In such cases, multiply the exponents: (a m ) n = a (mn) (3 2 ) 4 = 3 (2x 4) = 3 8 Multiplying and Dividing Numbers with Exponents

10. Special Exponents There are a few special properties of certain exponents that you also should know. Zero Any base raised to the power of zero is equal to 1. If you see any exponent of the form x 0 , you should know that its value is 1. Note, however, that 0 0 is undefined. One Any base raised to the power of one is equal to itself. For example, 2 1 = 2, (–67) 1 = –67, and x 1 = x . This can be helpful when you’re attempting an operation on exponential terms with the same base. For example: 3x 6 x x 4 = 3 x (6+4) = 3 x 7 Exponents- special cases