Sophia College Algebra 1. PERFORMING OPERATIONS WITH REAL NUMBERS | MILESTONE 1

Sophia College Algebra 1. PERFORMING OPERATIONS WITH REAL NUMBERS | MILESTONE 1 You passed this Milestone 18 questions were answered correctly. 3 questions were answered incorrectly. 1 The Elster family drove 9.25 hours on the first day of their road trip. How many minutes is this equivalent to? 9,256 minutes 33,300 minutes 555 minutes 154 minutes RATIONALE In general, we use conversion factors to convert from one unit to another. A conversion factor is a fraction with equal quantities in the numerator and denominator, but written with different units. We want to convert hours to minutes. We know how many minutes are in 1 hour. We will use this fact to set up a conversion factor. There are 60 minutes in 1 hour so to convert 9.25 hours into minutes, we will multiply by the fraction . Notice how the fractions are set up. The units of hours will cancel, leaving only minutes. Finally we can evaluate the multiplication by multiplying across the numerator and denominator. In the numerator, 9.25 times 60 equals 555. 9.25 hours is equivalent to 555 minutes. CONCEPT Converting Units 2 Perform the following operations and write the result as a single number. [4 + 8 × (5 – 3)] ÷ 5 + 6 10.8 10 2 1.8 RATIONALE Following the Order of Operations, we must first evaluate everything in parentheses and grouping symbols. When there are brackets or braces, evaluate the innermost operations first. Here, we must evaluate 5 minus 3 first. 5 minus 3 is 2. There are still operations inside grouping symbols to evaluate. Multiplication comes before addition, so we must evaluate 8 times 2 next. 8 times 2 is 16. Next, we add 4 and 16 to complete the operations inside parentheses. 4 plus 16 is 20. Now there is just division and subtraction. Division comes before subtraction in the Order of Operations, so we divide 20 by 5 next. 20 divided by 5 is 4. Lastly, add 4 and 6. 4 plus 6 is 10. CONCEPT Introduction to Order of Operations 3 Consider the following expression: What is the value of this expression when x = -5? • • • • RATIONALE To find the value of this expression when x = -5, begin by substituting -5 for every instance of x in the expression. Once all instances of x have been substituted with -5, we can evaluate the expression. The division bar acts as a group symbol, separating the expression in the numerator, |-5|, from the expression in the denominator, . Evaluate each of them separately before dividing, starting with the absolute value of -5. Recall that the absolute value of a number is its non-negative value. The absolute value of -5 is 5. Next, in the denominator, we can square -5. Remember that a negative number squared is always positive so -5 squared is 25. Finally, simplify the fraction. The fraction can be simplified to . CONCEPT Operations as Grouping Symbols 4 Simplify the following radical expression. • • • • RATIONALE To simplify this expression, we can use the Product Property of Radicals to separate the expression into two radicals. The cube root of can be written as the cube root of 64 times the cube root of . Next, we can write each radical expression using a fractional exponent in order to simplify. The index of the radical determines the denominator of the fractional exponent. The index here is 3, so each expression underneath the radical will be raised to the power. Now that we have changed our original expression from a radical to fractional exponents, we can evaluate and simplify the two expressions that are raised to the power. to the power of evaluates to 4 because 4 raised to the 3rd power is 64 ( ). To simplify , we multiply the two exponents together. 3 times equals 1 and is simply x. The expression simplifies to 4x. CONCEPT Applying the Properties of Radicals 5 What is the value of the following expression? -2 -6.4 RATIONALE When evaluating higher-order roots, it helps to break down the number underneath the radical into prime factors. -32 can be written as (-2)(-2)(-2)(-2)(-2). Notice that there are five factors of -2, and they are all underneath a fifth root. This means the expression underneath the radical simplifies to -2. can be simplified to -2. Lastly, we apply the negative sign in front of the radical. The original expression can be simplified to 2. CONCEPT Evaluating Radicals 6 Write the expression as a single power of b. • •