Title: Unit 11 Quiz
Course: Technical Math for Industry

Q: Evaluate: when
- 486 (Incorrect. This is the value of , and you should be finding the value of . That is, you should raise 2 to the fifth power and then multiply by 3.)
- 96 (Correct.)
- 30 (Incorrect. You may have correctly substituted: . However, it seems that you multiplied by 5 rather than using it as an exponent.)
- 7,776 (Incorrect. This expression is equal to . It appears that you did the multiplication before applying the exponent. The correct order of operations is to do exponents before multiplication.)
A: 96

Q: Evaluate if .
A: (a choice with no text)
If wrong: Incorrect. You may have thought that was equal to , but it is actually equal to 1.
If wrong: Incorrect. You probably made a mistake with the sign of the second term. , so
If right: Correct.
If wrong: Incorrect. You may have thought that was equal to 0, but it is actually equal to 1.

Q: Evaluate when .
- 32 (Incorrect. It seems that you overlooked the negative in the exponent. The second factor is equal to , which is equal to)
A: (a choice with no text)
If right: Correct.
If wrong: Incorrect. A negative exponent does not mean that the number is negative.The second factor is equal to , which is equal to
If wrong: Incorrect. You may have thought that was equal to 4, but it is actually equal to 1.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. In order to multiply the variable factors, you need to add the exponents, which you did. You also need to multiply the coefficients, not add them.
If right: Correct.
If wrong: Incorrect. You may have thought that the exponent of z is 0, or you may have known that it is 1 and thought you should multiply the exponents. Instead, you should apply the product rule, which says . That is, to multiply factors with like bases you add the exponents.
If wrong: Incorrect. Even though this is a multiplication problem, you do not multiply the exponents. Apply the product rule, which says . That is, to multiply factors with like bases you add the exponents. However, you need to multiply the coefficients.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. It seems that you added all of the exponents. There are two different rules of exponents that you must apply here. First you must simplify and using the power rule: . Then to complete the problem you must use the product rule: .
If wrong: Incorrect. It appears that you were thinking of the product rule, but you used the wrong order of operations. You must apply exponents before multiplication. So you must simplify and first. Use the power rule: . Then to complete the problem use the product rule: .
If right: Correct.
If wrong: Incorrect. It seems that you multiplied all of the exponents. There are two different rules of exponents that you must apply here. First you must simplify and using the power rule: . Then to complete the problem you must use the product rule: .

Q: Simplify:
A: (a choice with no text)
If right: Correct.
If wrong: Incorrect. When you simplified the fractions you probably divided exponents, but you should have subtracted them. Use the quotient rule, , to simplify each fraction. Then use the product rule.
If wrong: Incorrect. You may have correctly used the quotient rule twice to simplify the expression to . At this point you may have multiplied exponents when you should have added them.
If wrong: Incorrect. You may have divided exponents to get the incorrect result of , and then incorrectly simplified by multiplying the exponents. First use the quotient rule, , to simplify each fraction. Then use the product rule, .

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have mistakenly added exponents to simplify . The power rule tells you to multiply these exponents.
If wrong: Incorrect. You may have correctly applied the power rule and the product rule and simplified the expression to , but then divided the exponents instead of subtracting them.
If right: Correct.
If wrong: Incorrect. After correctly multiplying exponents to simplify to , you may have mistakenly multiplied exponents to simplify in the denominator. The product rule tells you to add these exponents.

Q: Simplify the expression:
A: (a choice with no text)
If wrong: Incorrect. Raising a negative number to an odd power means that you are multiplying an odd number of negative numbers. So the result will be negative.
If right: Correct. Apply the exponent 3 to each factor within the parentheses: Simplify, using the power rule:
If wrong: Incorrect. You need to raise every factor inside the parentheses to the power outside. You raised the variable factors to the third power, but you forgot to raise the constant to the third power.
If wrong: Incorrect. When you applied the exponent 3 to the factors and , you added the exponents instead of multiplying them.

Q: Simplify the expression.
A: (a choice with no text)
If wrong: Incorrect. A negative number raised to the third power is the same as the product of three negative numbers. The product of an odd number of negatives is negative. So is negative.
If wrong: Incorrect. You probably correctly applied the exponent 3 to the variable factors, used the Power Rule, and used the Quotient Rule. However, you should also have applied the exponent to the constant factor, .
If right: Correct. Apply the exponent 3 to each factor individually: Simplify and use the Power Rule: Use the Quotient Rule:
If wrong: Incorrect. You may have added the exponent 3 to the exponent of each variable factor instead of multiplying by it, and then used the Quotient Rule.

Q: Simplify the expression:
A: (a choice with no text)
If wrong: Incorrect. When multiplying and , you multiplied the exponents together instead of adding them.
If wrong: Incorrect. When multiplying the coefficients, you forgot that a positive number times a negative number is negative.
If right: Correct. Apply the exponent 3 to each factor in the first expression: Calculate and use the power rule twice: Multiply the coefficients and use the product rule:
If wrong: Incorrect. You probably applied the exponent 3 to the variable factors and correctly used the power and product rules. However, you forgot to apply the exponent 3 to the coefficient 5.

Q: Rewrite in decimal notation.
- 486,000,000,000,000 (Incorrect. You knew that the decimal should move to the right because the exponent is positive, and you chose an answer that has twelve zeros, perhaps because the exponent is 12. However, the exponent means that the decimal should be moved exactly 12 places.)
- 4.86 (Incorrect. You may have thought that you just had to drop the power of 10 after the decimal. However, you need to move the decimal point twelve places to the right to create a decimal equivalent to the number in scientific notation.)
- 0.00000000000486 (Incorrect. You moved the decimal point twelve places to the left. Because the exponent is positive, you need to move the decimal point twelve places to the right.)
- 4,860,000,000,000 (Correct. Since the exponent is positive, move the decimal point twelve places to the right.)
A: 4,860,000,000,000

Q: Divide, expressing the quotient in scientific notation.
A: (a choice with no text)
If wrong: Incorrect. You probably found the correct quotient , but when converting 0.48 to scientific notation you incorrectly wrote it as , instead of .
If right: Correct. Regroup: Divide the numbers: Use the quotient rule: Convert 0.48 into scientific notation: Regroup and use the product rule:
If wrong: Incorrect. This is the correct product, but it is not in scientific notation, because 0.48 is less than 1.
If wrong: Incorrect. You may have correctly regrouped and found the quotient of the numbers. However, when dividing the powers of 10 you found instead of .

Q: The volume of Jupiter is about cubic kilometers. The volume of Earth is about cubic kilometers. The number of Earths that can fit inside Jupiter can be found by dividing Jupiter's volume by Earth's volume. Find this quotient and express the answer in scientific notation.
- 1,300 (Incorrect. This is the correct quotient, but it is not expressed in scientific notation, because 1,300 is greater than 10 and there is no multiplication by a power of 10.)
A: (a choice with no text)
If right: Correct. Divide the volumes, regroup, and divide the numbers: Use the quotient rule:
If wrong: Incorrect. This is the correct quotient, but it is not expressed in scientific notation, because 13 is not between 1 and 10.
If wrong: Incorrect. You mistakenly multiplied the two volumes instead of dividing Jupiter's volume by Earth's volume.

Q: Which expression is not a polynomial?
- 12 (Incorrect. 12 is a monomial, so it is a polynomial.)
A: (a choice with no text)
If wrong: Incorrect. is the difference of two monomials, so it is a polynomial.
If right: Correct. is not a polynomial because this expression has a variable in the denominator. It is not the sum of monomials.
If wrong: Incorrect. is a monomial, so it is a polynomial.

Q: Evaluate for .
- 0 (Incorrect. You may have calculated by multiplying times 2 first. You need to evaluate exponents before multiplication.)
- 7 (Correct. Since , .)
A: 7
If wrong: Incorrect. You may have calculated by multiplying times 2 first and by multiplying times 2 first. You need to evaluate exponents before multiplication.
If wrong: Incorrect. You may have calculated by multiplying times 2 first. You need to evaluate exponents before multiplication.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have incorrectly simplified the term . The correct order of operations is to apply the exponent first and then multiply, so .
If wrong: Incorrect. You may have thought that you could combine unlike terms by adding coefficients. You can only combine terms with the same variable and exponent. The given polynomial equals , and you need to combine the like terms in the parentheses.
If wrong: Incorrect. You may have mistakenly thought that was equal to , when it is actually equal to 5.
If right: Correct. First identify the like terms. Then combine them by adding coefficients. Remember that .

Q: Find the sum:
A: (a choice with no text)
If wrong: Incorrect. You incorrectly tried to combine and , as well as and , but these are unlike terms and cannot be combined. Regroup and combine like terms to find the sum.
If wrong: Incorrect. You incorrectly tried to combine and , but these are unlike terms and cannot be combined. Regroup and combine like terms to find the sum.
If right: Correct. Rewrite subtraction as addition of the opposite: Regroup: Combine like terms:
If wrong: Incorrect. You ignored the minus signs when adding. You need to rewrite the problem as before regrouping and combining like terms.

Q: Find the opposite of .
A: (a choice with no text)
If wrong: Incorrect. You reversed the sign of only the first term in the polynomial, but each term must have its sign changed.
If wrong: Incorrect. You reversed only the positive signs in the polynomial, but each term must have its sign changed.
If right: Correct. Multiply the polynomial by : Distribute to each term in the polynomial: Multiply each coefficient by : Rewrite addition of a negative term as subtraction:
If wrong: Incorrect. You reversed only the negative signs in the polynomial, but each term must have its sign changed.

Q: Subtract:
A: (a choice with no text)
If wrong: Incorrect. You added the two polynomials instead of subtracting them. You need to change the subtraction to adding the opposite, making sure to change the sign of each term in the second polynomial. Then regroup and combine like terms.
If wrong: Incorrect. When changing the subtraction to addition of the opposite, you forgot to change the negative signs to positive in the second polynomial. You should add .
If right: Correct. Rewrite as adding the opposite: Regroup to match like terms: Combine like terms:
If wrong: Incorrect. When changing the subtraction to addition of the opposite, you forgot to change the positive signs to negative in the second polynomial. You should add .

Q: Find the area of the rectangle.
A: (a choice with no text)
If wrong: Incorrect. To find the area of a rectangle you must multiply the length by the width: . It appears that you multiplied the exponents of the monomials. In order to multiply monomials, multiply the coefficients and add the exponents.
If right: Correct. To find the area of a rectangle you must multiply the length by the width. In order to multiply monomials, multiply the coefficients and add the exponents.
If wrong: Incorrect. To find the area of a rectangle you must multiply the length by the width: . It appears that you added the coefficients of the monomials. In order to multiply monomials, multiply the coefficients and add the exponents.
If wrong: Incorrect. You may have tried to find the perimeter instead of finding the area. Also, you may have tried to add unlike terms by adding the coefficients and the exponents. To find the area of a rectangle you must multiply the length by the width. In order to multiply monomials, you need to multiply the coefficients and add the exponents.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. First, you must distribute the negative sign on the outside to every term of the polynomial. It was distributed only to the first term. Second, you must remember that y is the same as and that when you multiply monomials you must add the exponents.
If wrong: Incorrect. You must distribute the negative sign on the outside to every term of the polynomial. It was distributed only to the first term.
If wrong: Incorrect. To multiply a polynomial by a monomial, distribute the monomial to every term of the polynomial. It appears that you distributed only to the first term of the polynomial.
If right: Correct. To multiply a polynomial by a monomial, distribute the monomial to each of the terms in the polynomial. Remember to multiply coefficients and add exponents.

Q: Find the product:
A: (a choice with no text)
If right: Correct. Using the FOIL method, you find:
If wrong: Incorrect. It appears that you may have added the coefficients. Instead, you need to multiply each term in the first binomial with each term in the second binomial, and then combine the two like terms. FOIL is an easy way to remember that.
If wrong: Incorrect. Multiply each term in the first binomial with each term in the second binomial, and then combine the two like terms. It appears that you multiplied the first two terms and the last two terms, but and are missing.
If wrong: Incorrect. You may have correctly used FOIL, but then combined the middle terms incorrectly. If you are adding and , you must apply the same rules you learned for adding positive and negative numbers.

Q: Multiply:
A: (a choice with no text)
If wrong: Incorrect. It appears that you added the two polynomials together rather than multiplying them. Multiply each term in the first polynomial with each term in the second polynomial, and then combine the like terms.
If wrong: Incorrect. It appears that you ignored the negative sign on the 1. When you are multiplying individual terms, make sure that you carefully multiply the coefficients and find the right sign for each term.
If right: Correct.
If wrong: Incorrect. It appears that you did not distribute each term in the binomial to each term in the trinomial. Always multiply each term in the first polynomial with each term in the second polynomial.

Q: Square the binomial:
A: (a choice with no text)
If wrong: Incorrect. You left out the middle term. To square a binomial, you can use . In this case, and .
If right: Correct. . In this case, and , so .
If wrong: Incorrect. You have reversed the placement of the minus and plus signs. To square the binomial, you can use . In this case, and .
If wrong: Incorrect. You correctly squared each term, but the middle term, , is not correct. To square a binomial, you can use . In this case, and .

Q: Find the product:
A: (a choice with no text)
If wrong: Incorrect. You may have been thinking of the square of a binomial. To find this type of product, you can use . Note that there is no middle term in the result. In this case and .
If wrong: Incorrect. It appears that you doubled the terms instead of squaring them. To find this type of product, you can use . In this case and .
If right: Correct. To find the product, you can use . In this case and , so .
If wrong: Incorrect. It appears that you added the coefficients of the first terms instead of squaring them. To find this type of product, you can use . In this case and .

Q: A rectangle has an area of and a width of  . Find the length of the rectangle using the formula:
A: (a choice with no text)
If wrong: Incorrect. Using the given formula, the length is equal to . When dividing monomials, divide the coefficients and divide the variables by subtracting the exponents. You added the exponents.
If right: Correct. Using the given formula, the length is equal to . Simplify as follows:
If wrong: Incorrect. Using the given formula, the length is equal to . Divide the variables by subtracting the exponents. Because the x in the denominator did not have an exponent, you may have thought that you did not need to subtract anything. Remember that no exponent is the same as an exponent of 1.
If wrong: Incorrect. Using the given formula, the length is equal to . When dividing monomials, divide the coefficients and divide the variables by subtracting the exponents. You subtracted the coefficients.

Q: Divide:
A: (a choice with no text)
If wrong: Incorrect. Divide each term in the polynomial by the monomial. It appears that you divided only the middle term of the polynomial by the monomial.
If wrong: Incorrect. Divide each term in the polynomial by the monomial. It appears that when you divided the last term by , you subtracted the exponents incorrectly. The exponent is negative and must be simplified by moving the variable to the denominator of the fraction.
If right: Correct. Divide each term in the polynomial by the monomial.
If wrong: Incorrect. Divide each term in the polynomial by the monomial. It appears that you divided only the first term of the polynomial by the monomial.

Q: What is the remainder to the following polynomial division problem?
- 0 (Incorrect. During the first stage of the division process, when you should have subtracted , you may have added and . Then during the second stage you made a similar error in subtraction. Make sure that you distribute the minus sign when you get to the steps involving subtraction.)
- 12 (Correct.)
- 10 (Incorrect. During the first stage of the division process, when you should have subtracted , you may have added and . This would have given you left over for the next stage, when you should have had . Then on the second stage it appears that you correctly subtracted from the incorrect . Make sure that you distribute the minus sign when you get to the steps involving subtraction.)
A: 12
If wrong: Incorrect. It appears that you may have done the division process correctly up until the last step. At that point you should have subtracted , but you may have added 5 and instead. Make sure that you distribute the minus sign when you get to the steps involving subtraction.

Q: Divide.
A: (a choice with no text)
If wrong: Incorrect. During the first stage of the division process you should have subtracted from the dividend. While subtracting , you may have added the terms to the dividend. Then you made a similar mistake when you got to the next subtraction. Make sure that you distribute the minus sign when you get to the steps involving subtraction.
If wrong: Incorrect. It appears that you did not realize that the dividend did not have an term, and treated as if it were . When you set up a long division, first check to see if the dividend has any missing terms and put in a placeholder. In this case, the placeholder would be .
If right: Correct.
If wrong: Incorrect. It appears that you correctly applied the division process up until the last step. At that point you should have subtracted . You may have added the terms to . Make sure that you distribute the minus sign when you get to the steps involving subtraction.

Q: Evaluate the expression for and .
- 224 (Incorrect. You may have reversed the variables when substituting their values into the expression. The correct substitution is .)
- 104 (Correct.)
A: 104
If wrong: Incorrect. You may have incorrectly evaluated as 8 instead of .
If wrong: Incorrect. After simplifying the exponent, you may have evaluated the expression in order from left to right. Remember to follow the correct order of operations by multiplying before adding or subtracting.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You combined the like terms, but you added the exponents as well as the coefficients. When you combine like terms you are really using the distributive property, and this does not change the exponents.
If wrong: Incorrect. It appears that you incorrectly combined and to get , and then combined that with the other terms. When you combine like terms, the exponents do not change.
If right: Correct. To simplify an expression, reorder the addends and combine like terms.
If wrong: Incorrect. It appears that you added the coefficients of the like terms, ignoring the one negative sign. When you combine like terms and add their coefficients, make sure you use the rules you learned for adding positive and negative numbers.

Q: Add:
A: (a choice with no text)
If right: Correct. Remove the parentheses, rewrite subtraction as addition of the opposite, reorder, and combine like terms.
If wrong: Incorrect. You correctly added the coefficients of the like terms, but you also added their exponents. When you combine like terms, their exponents do not change.
If wrong: Incorrect. It appears that you added the first terms to each other and second terms to each other, even though they are not like terms. When adding polynomials, be sure to combine only like terms.
If wrong: Incorrect. You appear to have added all coefficients to make one term. When adding polynomials, be sure to combine only like terms.

Q: Subtract:
A: (a choice with no text)
If wrong: Incorrect. It appears that you inconsistently applied the minus sign to some terms in the second polynomial and not to others. When you subtract polynomials you should remove the parentheses and change the signs of all the terms of the second polynomial. Then combine the like terms.
If right: Correct. Remove the parentheses and change the signs of the terms of the second polynomial: Regroup and combine like terms:
If wrong: Incorrect. It appears that you added the polynomials instead of subtracting them. To subtract polynomials, remove the parentheses, change the signs of the terms of the second polynomial, regroup, and then combine like terms.
If wrong: Incorrect. It appears that you subtracted the first polynomial from the second polynomial. Perhaps you reversed them when lining them up vertically. To subtract polynomials, remove the parentheses, change the signs of the terms of the second polynomial, regroup, and then combine like terms.

Q: Multiply:
A: (a choice with no text)
If wrong: Incorrect. It appears that you multiplied the coefficients correctly, but then multiplied the exponents instead of adding them. To multiply a polynomial by a monomial, use the distributive property and the laws of exponents.
If wrong: Incorrect. It appears that you added the coefficients instead of multiplying them. To multiply a polynomial by a monomial, use the distributive property and the laws of exponents.
If right: Correct. To multiply a polynomial by a monomial, use the distributive property and the laws of exponents.
If wrong: Incorrect. It appears that you distributed only to the first term. You must distribute that factor to all terms in the parentheses.

Q: Divide:
A: (a choice with no text)
If right: Correct. To divide a polynomial by a monomial, divide each term separately by the denominator.
If wrong: Incorrect. You divided only the first term in the numerator by the denominator. To divide a polynomial by a monomial, divide each term separately by the denominator.
If wrong: Incorrect. You may have combined the terms in the numerator by subtracting the coefficients and adding the exponents and then dividing that result by the denominator. To divide a polynomial by a monomial, divide each term separately by the denominator.
If wrong: Incorrect. It appears that you subtracted the coefficients. When you divide variables, the exponents of the same variable can be subtracted. However, you should just do ordinary division with the coefficients.