Understanding Exponents and Absolute Value Concepts

18/21 that's 86% RETAKE 18 questions were answered correctly . 3 questions were answered incorrectly . 1 What is the following expression equivalent to?     RATIONALE First, apply the Power of a Power Property of Exponents, which states that when an exponent is another exponent, you can multiply the exponents. Therefore, multiply 2 and 4 to evaluate the . is equivalent to . Next, combine the two terms. The Product Property of Exponents st expressions with the same base are multiplied together, you can add the exponents. Add the e and 3 to evaluate the two terms. 8 plus 3 is 11, which becomes the final exponent. CONCEPT Properties of Exponents Report an issue with this question 2

Evaluate the expression by finding the absolute value. |10| – |‐5| + |‐3|  18  12  2  8 RATIONALE Evaluate each absolute value first. Absolute value is always non-negative. Rewrite the number negative sign if there was one. The absolute value of 10 is 10. The absolute value of -5 is 5. The absolute value of -3 is 3. Now and subtract these numbers following the order of operations. 10 minus 5 is 5. Next, we can add 3. 5 plus 3 is 8. CONCEPT Introduction to Absolute Value Report an issue with this question 3 Jayla bought the ingredients to make chicken soup, and wanted to make a double batch, which would be 12 cups of soup. A quick Google search told her that this was 173.3 cubic inches. She hoped the soup pot below would be big enough. The soup pot is 8 inches tall with a radius of 3 inches. What is the volume of the soup pot? Answer choices are rounded to the nearest whole cubic inch.  226 cubic inches  56 cubic inches

 603 cubic inches  150 cubic inches RATIONALE Recall that the volume of the pot can be represented with the formula for the volume of a cylin the value, r is the radius of the base, and h is the height. First, substitute 3 for r and 8 for h. Once we have the given values plugged into the appropriate places, we can evaluate the formu Order of Operations, we will first square the radius of 3. 3 squared equals 9. Next, we can multiply the remaining values. Using a calculator with a pi bu the most accurate; otherwise we can use the value 3.14. Multiplying π times 9 times 8 is 226, when rounded to the nearest whole cubic inch. The volume pot is 226 cubic inches. CONCEPT Volume Report an issue with this question 4 Consider the following expression: What is the value of this expression when x = -8?     RATIONALE To find the value of this expression when x = -8, begin by substituting -8 for every instance of x expression.

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Once all instances of x have been substituted with -8, we can evaluate the expression. The divi as a grouping symbol, separating the expression in the numerator, 2|-8|, from the expression in denominator, -8. Evaluate them separately before dividing, starting with evaluating the absolut Recall that the absolute value of a number is the non-negative value. The absolute value of -8 i the numerator, we can multiply 2 by 8. 2 times 8 is 16. Finally, divide 16 by -8. 16 divided by -8 is -2. CONCEPT Operations as Grouping Symbols Report an issue with this question 5 Perform the indicated operations and write your result as a single number.  39  43  63  91 RATIONALE For this expression, there is a lot to consider here. Follow the order of operations, and evaluate inside parentheses and other grouping symbols first. There are two groups. First, the radical sy (11 × 2 – 6), and a set of parentheses groups . Let's evaluate what is under the radical firs Under the radical, there is multiplication and subtraction. Multiplication comes before subtractio 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 ha the operations in the set of parentheses. Exponents comes before subtraction in the order of op 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 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 Report an issue with this question 6 Theresa bought a new desktop computer. One side of the desktop screen is 14 inches and the other side is 18 inches. What is the length of the diagonal of the desktop screen? Answer choices are rounded to the nearest inch.  20 inches  23 inches  16 inches  11 inches RATIONALE We can use the Pythagorean Theorem to calculate the length of a diagonal. The variables a and the sides of the computer, and c represents the diagonal. First, substitute 14 for a and 18 for b could also substitute 18 for a and 14 for b). Once we have the given values plugged into the Pythagorean Theorem, we can evaluate the ex 14 squared is 196, and 18 squared is 324. Now we can add these values together. 196 plus 324 is equal to 520. Finally, we can take the square root of both sides to find the value When we have a squared term, such as c², taking the square root of both sides will cancel this

The square root of 520 is approximately 23. The length of the diagonal, rounded to the nearest inches. CONCEPT Calculating Diagonals Report an issue with this question 7 Simplify the following radical expression.     RATIONALE To simplify this expression, we can use the Product Property of Radicals to separate the expres radicals. The cube root of can be written as the cube root of 27 times the cube root of . Next, w each radical expression using a fractional exponent in order to simplify. The index of the radica the denominator of the fractional exponent. The index here is 3, so each expression underneat will be raised to the power. Now that we have changed our original expression from a radical to fractional exponents, we ca and simplify the two expressions that are raised to the power. 27 to the power of evaluates to 3 because 3 raised to the 3rd power is 27 ( ). To simpl multiply the two exponents together. 3 times equals 1 and is simply x. The expression simplifies to 3x. CONCEPT

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Applying the Properties of Radicals Report an issue with this question 8 Simplify the following radical expression.     RATIONALE To simplify this expression, we can rewrite 180 into products of smaller numbers. There are ma this, but it can help to use a perfect square, since they simplify to integers when we evaluate th 36 is a perfect square and it is also a factor of 180. We can rewrite 180 as 36 times 5. Now we Product Property of Radicals to write the factors as separate radicals. The Product Property allows us to write the radical as the product of two individual square roots can evaluate the square roots. The square root of 36 is 6. The square root of 5 cannot be simplified further. The fully simplified . CONCEPT Simplifying Radical Expressions Report an issue with this question 9 What is the value of the following expression?  -6.4

 2  -2  6.4 RATIONALE When evaluating higher-order roots, it helps to break down the number underneath the radical factors. -32 can be written as (-2)(-2)(-2)(-2)(-2). Notice that there are five factors of -2, and they are al 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 Report an issue with this question 10 Find the value of this expression.  -13  17  19  -11 RATIONALE To solve this expression, evaluate the exponent for each term. Start with the first term, . Any number taken to the power of zero equals 1, so is equal to 1. Evaluate the next term,

When the exponent is 1, the value of the term is the same as its base, so remains 3. Next term , which is the same as (-4)(-4). Negative 4 times negative 4 equals positive 16. The last term, indicates that -1 is multiplie three times. is equivalent to -1, because when a negative number is multiplied by itself an odd number answer remains negative. Finally, evaluate the addition and subtraction from left to right, starti 3. -1 plus 3 is 2. Next, add 2 and 16. 2 plus 16 is 18. Finally, evaluate 18 minus -1. 18 minus -1 is the same as 18 plus 1, or 19. CONCEPT Introduction to Exponents Report an issue with this question 11 Which of the following equations is correctly calculated?  -7 × -7 = -49  -8 × 3 = -24  30 ÷ 3 = -10  -18 ÷ 6 = 3 RATIONALE This is correct. The product of a positive and a negative number is always negative. This is incorrect. The product of two negative numbers is always a positive number. The correc 49. This is incorrect. The quotient of two positive numbers is always positive. The correct quotient

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This is incorrect. The quotient of a positive and a negative number is always negative. The cor -3. CONCEPT Multiplying and Dividing Positive and Negative Numbers Report an issue with this question 12 How can the following expression be simplified and written without negative exponents?     RATIONALE By completing a series of steps, this expression can be written so that no negative exponents a The first step is to write the numerator as a single power of a. You can combine the two terms b Product Property of Exponents, which states that if two expressions with the same base are mu together, you can add the exponents. Therefore, add the exponents 7 and -22 to evaluate the t terms. 7 plus -22 equals -15, which becomes the new exponent in the numerator. Next, divide the num denominator and write this as a single power of a. To do this, use the Quotient Property of Expo says that when you divide two expressions with the same base, you can subtract the exponent evaluate -15 minus -10. Be careful when subtracting negative numbers! -15 minus -10 can be thought of as -15 plus 10 -15 plus 10 is equal to -5. This is the new exponent for a. Lastly, write this without any negative Write the expression as a fraction with 1 in the numerator, and change the sign of the exponen negative to positive. This is the simplified expression without any negative exponents. CONCEPT

Negative Exponents Report an issue with this question 13 Tom owns a square plot of land. He knows that the area of the plot is between 3000 and 3100 square meters. Which of the following is a possible value for the side length of the plot of land?  30 meters  100 meters  85 meters  55 meters RATIONALE Tom owns a plot of land that is perfectly square. The area of the plot is between 3000 and 3100 meters. One way to solve this problem is to find the area for each side length given. The area o special case in which the length and the width are the same, so you can find the area by squar If the side length of the square is 30 meters, then the area equals 900 square meters. This is N length for Tom’s plot because 900 is not between 3000 and 3100. If the side length of the square is 85 meters, then the area equals 7225 square meters. This is N length for Tom’s plot because 7225 is not between 3000 and 3100. If the side length of the square is 100 meters, then the area equals 10000 square meters. This possible length for Tom’s plot because 10000 is not between 3000 and 3100. If the side length of the square is 55 meters, then the area equals 3025 square meters. This IS length for Tom’s plot because 3025 is between 3000 and 3100. CONCEPT Area Report an issue with this question 14 Write the expression as a single power of b.

    RATIONALE Start by simplifying the terms in the parentheses. Using the Quotient Property of Exponents, di terms that have the same base by subtracting the exponents, and . When subtracting fractions with a common denominator, subtract across the numerators and le denominator the same. minus equals . Next, apply the Power of Property of Exponents to multiply the tw and write b as a single power. times equals . Lastly, rewrite the fraction in its simplest form. simplifies to , so the expression simplifies to . CONCEPT Properties of Fractional and Negative Exponents Report an issue with this question 15 The diameter of a proton is about meters. A hydrogen atom has an overall length of 100,000 times (or times) the diameter of a proton. What is the length of the hydrogen atom, in meters, if it were written in scientific notation?  meters

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 meters  meters  meters RATIONALE To find the length of the hydrogen atom, multiply by 100,000, which can be expressed as When multiplying numbers in scientific notation, you must deal with the numbers and 10s sepa multiply 1.9 and 1. 1.9 times 1 equals 1.9. Now you can use the Product Property of Exponents on the 10s and add exponents. -15 plus 5 equals -10, which is the exponent for the base 10. Finally, combine the number part of 10 together. The length of the hydrogen atom (in meters), when written in scientific notation, is . CONCEPT Multiplication and Division in Scientific Notation Report an issue with this question 16 The speed of light is approximately 299,000,000 meters per second. What is the speed of the light, in meters per second, using scientific notation?  meters per second  meters per second  meters per second  meters per second

RATIONALE To express this number in scientific notation, you must find two parts: a number between one a be equal to one, but not ten), and a power of 10. To find the first part, place the decimal after t zero number, 2. The decimal will be placed after the 2. We will also include the numbers right after the 2, which These will go immediately after the decimal. To find the second part, you must find the expone will be raised. To do this, count how many times the decimal is moved from 2.99 to obtain 299, If we start with the decimal right after the 2, it would take us eight moves to the right to obtain number. Because we must move the decimal point eight times, the exponent is 8. Combine the two parts to state the solution in scientific notation. The speed of light is approxim meters per second. CONCEPT Writing Numbers in Scientific Notation Report an issue with this question 17 Tyler spent 17,700 seconds completing a 1,000 piece puzzle. How many hours is this equivalent to? Answer choices are rounded to the nearest hour.  288 hours  5 hours  20 hours  124 hours RATIONALE In general, we use conversion factors to convert from one unit to another. A conversion factor i with equal quantities in the numerator and denominator, but written with different units. We wa seconds to hours. We might not know off-hand how many seconds are in an hour, but we know seconds are in a minute, and how many minutes are in an hour. We will use these facts to set u factors. There are 60 minutes in 1 hour and 60 seconds in 1 minute. To convert 17700 seconds into hou multiply by the fractions and .

Notice how the fractions are set up. The units of seconds and minutes will cancel, leaving only we can evaluate the multiplication by multiplying across the numerator and denominator. In the denominator, 60 times 60 equals 3600. 17700 divided by 3600 is approximately 4.92. R nearest hour, 17700 seconds is equivalent to 5 hours. CONCEPT Converting Units Report an issue with this question 18 Perform the following operations and write the result as a single number. 34 – 18 – (-23)  75  -7  39  29 RATIONALE We can first evaluate the subtraction from left to right. Begin by evaluating 34 minus 18. 34 minus 18 equals 16. Next evaluate 16 minus negative 23. Subtracting a negative is equivale its opposite: so 16 – (-23) is the same as 16 + 23. 16 plus 23 equals 39. CONCEPT Adding and Subtracting Positive and Negative Numbers Report an issue with this question 19 Perform the following operations and write the result as a single number. [4 + 8 × (5 – 3)] ÷ 5 + 6  1.8  10.8

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 10  2 RATIONALE Following the Order of Operations, we must first evaluate everything in parentheses and group When there are brackets or braces, evaluate the innermost operations first. Here, we must eva 3 first. 5 minus 3 is 2. There are still operations inside grouping symbols to evaluate. Multiplication com 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 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 Report an issue with this question 20 Simplify the following expression in terms of fractional exponents and write it in the form .    

RATIONALE Radical expressions can be rewritten using fractional exponents by corresponding the index of the denominator of the fractional exponent. Begin by writing the expression underneath the rad entire expression will be raised to a fractional exponent power. The index of the radical is 3. In other words, this is the third-root. This means that the entire ex be raised to the power of . Next, to write this in the form , distribute the outside expon the powers of 10 and x. Once the exponent of is distributed to both terms, we can simplify by multiplying the expo and for the term of 10. In this case, becomes . The final expression is . CONCEPT Fractional Exponents and Radicals Report an issue with this question 21 Consider the following set of real numbers: Which of the following contains ALL of the irrational numbers in the set?     RATIONALE

Rational numbers can be expressed as a ratio of two integers, and are characterized by either t repeating decimal patterns, such as 0.375 or 0.3333... Irrational numbers are characterized by terminating, non-repeating decimal pattern. Evaluate each number and determine whether it is irrational. Irrational: pi has a non-terminating, non-repeating decimal pattern. Irrational: evaluates to -1.7320508... It has a non-terminating, non-repeating decimal patt Rational: The digits terminate after the 5 in the tenths place. Rational: The digits terminate after 0. Rational: This is a ratio of two integers, 1 and 7. Irrational: This evaluates to 2.2360679... It has a non-terminating, non-repeating decimal patte Rational: The square root of 9 evaluates to the integer 3. Rational: This has a repeating decimal pattern. The irrational numbers in the set are .

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