Sophia College Algebra MM212 Final Milestone 20/25 that's 80%

Sophia College Algebra MM212 Final Milestone 20/25 that's 80% 20 questions were answered correctly. 5 questions were answered incorrectly. 1 Geri runs a total of 3.1 miles in a cross-country race. How many feet is this equivalent to? • 5,280 feet • 16,368 feet • 4,989 feet • 15,840 feet 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 miles to feet. We know how many feet are in 1 mile. We will use this fact to set up a conversion factor. There are 5280 feet in 1 mile. To convert 3.1 miles into feet, multiply by the fraction . Notice how the fractions are set up. The units of miles will cancel, leaving only feet. Finally we can evaluate the multiplication by multiplying across the numerator and denominator. In the numerator, 3.1 times 5280 equals 16368. 3.1 miles is equivalent to 16,368 feet. CONCEPT Converting Units 2 Perform the indicated operations and write your result as a single number. R ATIONALE For this expression, there is a lot to consider here. Follow the order of operations, and evaluate anything inside parentheses and other grouping symbols first. There are two groups. First, the radical symbol groups (11 × 2 – 6), and a set of parentheses groups . Let's evaluate what is under the radical first. Under the radical, there is multiplication and subtraction. Multiplication comes before subtraction in the order of operations, so we multiply 11 by 2 to get 22. Next, evaluate the subtraction. 22 minus 6 is 16. Now we can take the square root of 16. The square root of 16 is 4. This is the simplified expression underneath the radical. Next, we have to evaluate the operations in the set of parentheses. Exponents comes before subtraction in the order of operations, so we square 4 first. 4 squared equals 16. Next, we subtract 9 from 16. 16 minus 9 equals 7. Now that we have eliminated the grouping symbols, we can evaluate the multiplication of 5 and 7. 5 times 7 equals 35. Finally, we can add 4 and 35. 4 plus 35 is equal to 39. The expression simplifies to 39. CONCEPT Order of Operations: Exponents and Radicals 3 RATIONALE When given a rectangle with unknown measurements, we can find the area of the rectangle by multiplying the length and the width. We need to express the length and width by adding the partial dimensions of each side. The length is , and the width is . Next, we can find the area by using binomial multiplication. The area is equivalent to length times width, or times . When multiplying two binomials, we can use FOIL. Multiply the first terms , the outside terms , the inside terms , and the last terms . Once we have FOILed, we can evaluate each multiplication. The expression multiplies to Next, combine like terms and . Once all like terms have been combined, write the expression in standard form by decreasing degree. The area of the rectangle is . CONCEPT Using FOIL to Represent Area 4 RATIONALE This question involves several properties of logarithms. The Quotient Property of Logs states that division inside a logarithm can be expressed as subtraction of individual logarithms. This means we can express as . Next, the Product Property of Logs states that multiplication inside a logarithm can be expressed as addition of individual logarithms. This means we can express as . Then, the Power Property of Logs states that exponents inside a logarithm can be expressed as outside scalar multiples of the logarithm. This means we can express as and as . Finally, we can substitute the values we were previously given for , , and . Recall that , , and . Once these given values are substituted into the expression, simplify each term and then perform the addition and subtraction. times is ; and times is . The logarithmic expression evaluates to . CONCEPT Applying Properties of Logarithms 5 RATIONA LE Radical expressions can easily be added and subtracted similar to combining like terms, if there is the same number underneath the radical. If they are different, use the Product Property of Radicals to simplify if possible. Let's simplify the first term, . can be written as because 4 times 5 is 20. We can simplify the square root of 4 further. The square root of 4 is 2, so the expression becomes when fully evaluated. Let's consider the next radical, . is already in its simplest form, since 5 is a prime number that cannot be broken down any further. Let's take a look at the third term, . can be written as because 9 times 5 is 45. We can simplify the square root of 9 further. The square root of 9 is 3, so the expression becomes when fully evaluated. Now we can rewrite the original expression with the newly evaluated terms. can be rewritten as . We have simplified the expression to radical expressions with the same number underneath the radical sign. Lastly, we can combine all of the coefficients in front of the radical sign. 2 minus 3 plus 6 equals 5. The expression is equivalent to . CONCEPT Adding and Subtracting Radical Expressions 6