Title: Unit 16 Quiz
Course: Technical Math for Industry

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have chosen this answer because . However, , using the rule . Since is not equal to 8, .
If wrong: Incorrect. You correctly recognized that you can rewrite . Therefore, . However, this is not equal to . You need to use to rewrite the expression as , and then simplify further.
If wrong: Incorrect. You may have correctly factored as . But while taking the square root of , you mixed up the places of 5 and 4.
If right: Correct. Factor 80 so that you have a perfect square times a number with no square factors: Use to rewrite the expression: Take the square root of :

Q: Approximate to the tenths place.
- 8.5 (Incorrect. You may have chosen this approximation because it falls between 8 and 9 and you know that is between and . However, because 76 is closer to 81 than it is to 64, the approximation will be closer to 9 than to 8. Using trial and error, square different approximations to find the best estimate of .)
- 8.8 (Incorrect. This is a reasonable approximation, but it is not the best approximation to the tenths place. Compare to squares of other decimals between 8 and 9.)
- 8.7 (Correct. 76 is between the perfect squares 64 and 81. So is between and . and Because 76 is closer to 81 than it is to 64, is probably about 8.7 or 8.8. and 76 is closer to 75.69 than 77.44, so the answer is 8.7.)
A: 8.7
If wrong: Incorrect. This answer is equal to but the problem asked to approximate to the tenths place. Therefore, your answer should be in decimal form.

Q: Simplify:
A: (a choice with no text)
If right: Correct. Factor to find variables with even exponents: Rewrite those as squares : Separate the squares into individual radicals: Find square roots. Use absolute value for odd exponents: Simplify:
If wrong: Incorrect. You took the square roots of the factored variables, but left them under the radical sign. You also took out the remaining factors that should have stayed within the radical and placed them outside within absolute value symbols.
If wrong: Incorrect. You may have tried to take the square roots of the exponents of each variable. Actually, when you compute a square root you end up with half of the exponent. Factor to find variables with even exponents and rewrite those as perfect squares. Then use and simplify.
If wrong: Incorrect. You forgot to take the absolute value of . Because the variable has an odd exponent, the expression could have a positive or negative value. You need to write because the square root symbol means take the positive root.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You found three times the square root of the radical expression instead of finding the cube root of the radical expression. Factor out perfect cubes from the constant and the variables. Rewrite as a product of radicals and simplify.
If wrong: Incorrect. This is equivalent to the original expression, but it is not in simplest form. Rewrite as . Then split the expression into a product of radicals and simplify.
If right: Correct. Factor 250 to find a perfect cube: Factor to find a perfect cube : Rewrite as a product of radicals: Simplify and multiply:
If wrong: Incorrect. You found the square root instead of the cube root. Identify cubed numbers and variables. Rewrite as a product of radicals and simplify.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have correctly separated the factors into perfect squares and then taken the square roots of each. However, you left out the factor when multiplying and simplifying. You also forgot the rule that if n is even, .
If wrong: Incorrect. You correctly separated the factors into perfect squares and took the square roots of each, but you forgot the rule that if n is even, .
If right: Correct. Factor out perfect squares: Separate the factors into individual radicals: Find the square roots: Simplify:
If wrong: Incorrect. You correctly separated the factors into perfect squares and took the square roots of each, but you left out the factor when multiplying and simplifying.

Q: Write as an expression with a rational exponent.
A: (a choice with no text)
If right: Correct. Any radical in the form can be rewritten as . So
If wrong: Incorrect. You were probably thinking about , which could be rewritten as . Any radical in the form can be rewritten as .
If wrong: Incorrect. Any radical in the form can be rewritten as . You reversed the placement of the index and the exponent from the radical.
If wrong: Incorrect. You were probably thinking about , which could be rewritten as . Any radical in the form can be rewritten as .

Q: Express in radical form.
A: (a choice with no text)
If right: Correct. The rule for rewriting rational exponents is: The denominator of the fraction is 4, so this denominator becomes a fourth root. Because there are no parentheses, the exponent refers only to the quantity immediately to its left.
If wrong: Incorrect. You reversed the placement of the index and the exponent. The rule for rewriting rational exponents is:
If wrong: Incorrect. Because has no parentheses, the exponent refers only to the quantity immediately to its left. So the factor of 7 will be outside of the radical.
If wrong: Incorrect. Because has no parentheses, the exponent refers only to the quantity immediately to its left. So the factor of 7 will be outside of the radical.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You changed each expression in the fraction to radical form, but this is not the simplest form of the expression. The correct way to simplify would be to use the rule , then convert to radical form.
If right: Correct. Use the rule : Change to radical form:
If wrong: Incorrect. You may have been thinking about division of fractions. You may have multiplied the exponent by to get and then , but you should have used the rule:
If wrong: Incorrect. You may have tried to simplify , when you should have been simplifying instead.

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have rewritten the radical using a rational exponent, but you incorrectly factored 1,000 and rewrote it as instead of before simplifying.
If right: Correct. Rewrite the radical using a rational exponent: Use the laws of exponents to simplify: Factor 1,000 and simplify: Change the factors with exponent back to radical form:
If wrong: Incorrect. You may have correctly rewritten the radical as , but you did not use the laws of exponents to simplify. You applied the exponent only to the last factor, . Instead, you should rewrite this expression as , rewrite 1,000 in exponential form, and simplify the exponents.
If wrong: Incorrect. You may have correctly rewritten the radical as , but you did not use the laws of exponents to simplify. You used 6 as the numerator and each factor's exponent as a denominator before simplifying. You should have multiplied each factor's exponent by instead.

Q: Which one of the following simplifications is incorrect?
A: (a choice with no text)
If wrong: Incorrect. This is a correct simplification because:
If right: Correct. The first radical is a fourth root and the second radical is a square root. You cannot multiply the product of the radicands and put it under the same radical sign.
If wrong: Incorrect. This is a correct simplification because:
If wrong: Incorrect. This is a correct simplification because:

Q: Divide and simplify:
A: (a choice with no text)
If wrong: Incorrect. You found an equivalent expression, but this expression will simplify further. Identify perfect squares and pull them out of the radical.
If wrong: Incorrect. You may have simplified by subtracting the coefficients and then simplified correctly. Rewrite using and simplify by identifying similar factors in the numerator and denominator. Be sure to divide the coefficients and subtract the exponents. Identify perfect squares and pull them out of the radical.
If wrong: Incorrect. You may have ignored the radical signs or incorrectly thought that they canceled. Rewrite using and simplify. Pull out perfect squares from and simplify, using .
If right: Correct.

Q: Add and simplify:
A: (a choice with no text)
If wrong: Incorrect. You may not have noticed the index of the cube root, and so incorrectly combined the first two radicals, while not simplifying the third one. Start by simplifying the last term using . Now the first and last terms are like terms that you can combine, just as you would combine .
If wrong: Incorrect. You may have simplified the third radical and then tried to combine all three terms. After correctly simplifying, you can combine the first and third terms. This is because they are both multiples of , which makes them like terms. The second term is a multiple of , so it is an unlike term.
If right: Correct. Simplify the last term using : The first and third terms are like terms and can be combined: So the entire expression equals . Because the indices are not the same, this cannot be further simplified.
If wrong: Incorrect. You may have mistaken for , and then added the first and third terms. Start by simplifying the last term using . Now the first and third terms are like terms that you can combine, just as you would combine .

Q: Subtract and simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have simplified the radicals correctly, but then tried to cancel a factor of with a term . You can only combine like terms by addition or subtraction. If one radical has a variable coefficient and the other does not, then they are not like terms.
If wrong: Incorrect. You may have tried to combine the first and third terms by putting the subtraction under one radical sign, then simplifying further. When you multiply or divide radicals you can combine them under one sign, but cannot be combined as . Instead, simplify the first and third terms using . Then you will end up with like terms that can be combined.
If wrong: Incorrect. This is equivalent to the original expression but is not in simplest form. Rewrite the first term in this answer using , and then simplify.
If right: Correct. Simplify each radical and combine like terms. This could also be written as .

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. It appears that you used the Distributive Property correctly, but this expression is not in simplest form. Rewrite each radical using , pulling out all perfect cubes. Then combine any like terms.
If wrong: Incorrect. It appears that you distributed correctly, but did not simplify the expression all the way. Be sure to combine all like terms.
If right: Correct. Use the Distributive Property:
If wrong: Incorrect. It appears that you multiplied only the first term inside the parentheses by , and then simplified each term correctly. Make sure to apply the Distributive Property first. Then simplify each term, using , and combine like terms.

Q: Rationalize the denominator and simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have thought that you could rewrite the numerator as , then proceeded to rationalize the denominator. You may have been thinking of , but there is no similar rule for addition. The first two steps are: Multiply the radicals, then simplify each term.
If wrong: Incorrect. You may have found the correct answer, but then simplified incorrectly. You may have thought that the coefficients (a and b) of the numerator terms and the ab in the denominator reduced to 1. However, a and b are not factors of the numerator, so this kind of simplification cannot be done.
If wrong: Incorrect. You didn't distribute when you multiplied the numerator by . The first two steps are: Multiply the radicals, then simplify each term.
If right: Correct. Rationalize the denominator by multiplying the expression by:

Q: Rationalize the denominator and simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have found the correct answer, but then thought that the 64 on the top and the bottom would simplify to 1. That kind of simplification would hold only if 64 were a factor of the numerator and denominator.
If right: Correct. Multiply the numerator and denominator by the conjugate of the denominator:
If wrong: Incorrect. You did not distribute when multiplying the numerator by the denominator's conjugate. Multiply the numerator and denominator by the conjugate of the denominator: Use the Distributive Property and FOIL and simplify radicals.
If wrong: Incorrect. You may have thought that the 8 on the top and the bottom would simplify to 1. That kind of simplification would hold only if 8 were a factor of the denominator. Multiply the numerator and denominator by the conjugate of the denominator: Use the Distributive Property and FOIL and simplify radicals.

Q: Which equation has as a solution?
A: (a choice with no text)
If wrong: Incorrect. You may have chosen this equation because its solution is . However, when you substitute into the left side you get , which is not a real number.
If wrong: Incorrect. You may have chosen this equation because its solution is . However, when you substitute into the left side you get , which is not a real number.
If right: Correct. , so
If wrong: Incorrect. You may have substituted into the equation, got 9 under the radical, and forgot to take the square root. The left side would actually be equal to 3, so it is not a solution of this equation.

Q: Solve:
- There is no solution. (Correct. Square both sides: Simplify: Solve for x: Check: ? | | extraneous solution Therefore, there is no solution.)
A: There is no solution.
If wrong: Incorrect. You may have squared the left side and not the right side, to get the incorrect equation . That equation would have as a solution. Make sure that you always perform the same operation on both sides of an equation. In this case you will get . Solve this equation and check the solution back in the original equation.
If wrong: Incorrect. You may have squared both sides and solved the resulting equation correctly, but did not check the solution in the original equation. You need to substitute into the given equation to see if it is an actual solution or an extraneous solution.
If wrong: Incorrect. You may have tried to square both sides of the equation, but incorrectly got on the right side. Remember that the square of a negative number is a positive number. Also, be sure to check all solutions by substituting them back into the original equation. If you get a false statement, then the solution is extraneous.

Q: The period T (in seconds) of a pendulum is given by , where L stands for the length (in feet) of the pendulum. If , and the period is 3.14 seconds, what is the length?
- 8 feet (Correct. , so , so)
- 80 feet (Incorrect. You may have correctly started with and simplified to , but when you squared both sides you misplaced a decimal point.)
- 32 feet (Incorrect. You may have correctly started with and simplified to , but when you squared the right side, you doubled the left side. When solving any equation, make sure that you perform the same operation on both sides.)
- 16 feet (Incorrect. You may have correctly started with and simplified to , but when you squared the right side, you forgot to square the left side. When solving any equation, make sure that you perform the same operation on both sides.)
A: 8 feet

Q: Simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have found the correct answer, but then tried to simplify further by multiplying the number outside the radical by the number under the radical, which is not correct. Use the rule and the definition to simplify.
If wrong: Incorrect. You may have correctly noticed the perfect square 36 as a factor of 108, and correctly used , but you forgot the remaining factor of , which is 3. Use the rule and make sure that you write down every step.
If right: Correct. Use the rule and the definition of i.
If wrong: Incorrect. You may have used as the square root of , but that is not correct because does not equal . Use the rule and the definition to simplify.

Q: Identify the imaginary part of this complex number:
A: (a choice with no text)
If wrong: Incorrect. A complex number is a number in the form where a and b are real numbers. The term is referred to as the imaginary part. The number by itself is just the coefficient of the imaginary part.
If wrong: Incorrect. You may have chosen this answer because i is imaginary, but i is not the imaginary part. A complex number is a number in the form where a and b are real numbers. The term is referred to as the imaginary part.
If right: Correct. A complex number is a number in the form where a and b are real numbers. The term is referred to as the imaginary part. So is the imaginary part.
If wrong: Incorrect. You may have reversed the real and the imaginary parts of the complex number. A complex number is a number in the form where a and b are real numbers. The term is referred to as the imaginary part.

Q: Add:
A: (a choice with no text)
If wrong: Incorrect. It appears that you ignored the negative signs when adding. Rearrange the sum to put like terms together. Pay attention to the positive and negative signs when combining those like terms.
If wrong: Incorrect. You may have applied the negative sign in the second complex number to the term . The two imaginary parts that you should be combining are and .
If wrong: Incorrect. You may have found the correct answer but then tried to combine the real part with the imaginary part. However, these are not like terms.
If right: Correct. Rearrange the sum to put like terms together, and then combine those like terms.

Q: Subtract:
- 3 (Correct. Distribute the subtraction sign to both terms in the subtrahend. Rearrange the terms to put like terms together, and then combine like terms.)
A: 3
If wrong: Incorrect. It appears that you did not distribute the minus sign to the term . Distribute the subtraction sign to both terms in the subtrahend. Rearrange the terms to put like terms together, and then combine like terms.
If wrong: Incorrect. It appears that you added instead of subtracting. Distribute the subtraction sign to both terms in the subtrahend. Rearrange the terms to put like terms together, and then combine like terms.
If wrong: Incorrect. You may have incorrectly combined with 6. Distribute the subtraction sign to both terms in the subtrahend. Rearrange the terms to put like terms together, and then combine like terms.

Q: Multiply and simplify:
A: (a choice with no text)
If wrong: Incorrect. You found a correct product, but it is not in simplest form. Remember that is equal to .
If right: Correct. Multiply the coefficients of i together, and remember that .
If wrong: Incorrect. It appears that you multiplied the coefficients but not the i's. Use the fact that .
If wrong: Incorrect. It appears that you added the coefficients. Multiply the coefficients of i together, and remember that .

Q: Multiply and simplify:
- 58 (Incorrect. You may have thought that the problem showed a complex number being multiplied by its conjugate, so you added 49 to 9. Be sure to use FOIL and then simplify.)
A: (a choice with no text)
If wrong: Incorrect. You may have simplified to , but , because is equal to .
If right: Correct. Use FOIL and simplify.
If wrong: Incorrect. You may have found the correct answer, and then tried to combine the real and the imaginary part to make one term. The real part and the imaginary part are unlike, so they cannot be combined.

Q: Simplify:
A: (a choice with no text)
If right: Correct. Treat the division as a fraction, simplify, and then rationalize the denominator.
If wrong: Incorrect. You have found an equivalent expression. However there is an i in the denominator of the first term, so it must be rationalized. Multiply by and simplify.
If wrong: Incorrect. You have found an equivalent expression. However, there is an i in the denominator, so it must be rationalized. Multiply by and simplify.
If wrong: Incorrect. You may have simplified correctly and multiplied by to rationalize the denominator. At that point you either simplified the denominator incorrectly or omitted it. Remember that .

Q: Solve:
A: (a choice with no text)
If wrong: Incorrect. It appears that you used the wrong order of operations and solved the equation . The correct first step is to subtract 5 from both sides to get . Take the square root of both sides, isolate x, and simplify the radical.
If wrong: Incorrect. This is one solution to the equation, but the equation has two solutions. Also, the radical is not simplified.
If wrong: Incorrect. You probably did the first two steps correctly, but when you got to isolating the variable, you subtracted when you should have added 5 to both sides. Also, you need to simplify the radical.
If right: Correct. Subtract 5 from both sides: Use the Square Root Property: Add 5 to both sides: Simplify the radical:

Q: Find the number to add to to make it a perfect square trinomial. Write that trinomial as the square of a binomial.
- add 196; (Incorrect. You would add 196 to to make it . If you start with , add the number to complete the square. The resulting trinomial will be Here .)
- add 28; (Incorrect. is not equal to . If you start with , add the number to complete the square. The resulting trinomial will be Here .)
- add 14; (Incorrect. is not equal to . If you start with , add the number to complete the square. The resulting trinomial will be Here .)
- add 49; (Correct. If you start with , add the number to complete the square. , so add The resulting trinomial is)
A: add 49;

Q: Solve:
A: (a choice with no text)
If wrong: Incorrect. You may have correctly rewritten the equation as , but when you used the Square Root Property, you forgot that there are two square roots. You also need to simplify the radical.
If wrong: Incorrect. You may have correctly rewritten the equation as , but made a mistake with signs at some point. You also need to simplify the radical.
If wrong: Incorrect. You may have correctly rewritten the equation as , but made a mistake with signs at some point.
If right: Correct. Add to both sides to complete the square: Write the left side as the square of a binomial: Take the square root of both sides: Solve for x:

Q: Rewrite the equation in standard form and identify a, b, and c.
A: (a choice with no text)
If wrong: Incorrect. Standard form of a quadratic equation is , so you must rewrite the equation with one side equal to 0. You correctly reordered the terms on the left side, but forgot to subtract from both sides.
If wrong: Incorrect. It appears that you rewrote the equation so that the right side was 0, and correctly reordered the terms, but misread a coefficient. Do not forget to include any minus signs that may appear.
If wrong: Incorrect. It appears that you simply read the three constants from left to right. Standard form of a quadratic equation is . You must rewrite the equation so that one side is 0. Then reorder the terms to match this form and find a, b, and c.
If right: Correct. Standard form of a quadratic equation is . Rewrite so that the right side is 0: Reorder the terms:

Q: Use the Quadratic Formula to solve the equation .
- or (Correct. First rewrite as . Then use the Quadratic Formula, , with , and :)
- or (Incorrect. You probably correctly rewrote the equation as and used the Quadratic Formula, , but made a mistake when finding .)
- or (Incorrect. You probably correctly rewrote the equation as and used the Quadratic Formula, , but divided by 2 instead of .)
- or (Incorrect. You may have rewritten the equation as , instead of in standard form, and then substituted in the wrong values for b and c. You should be using , and in the formula:)
A: or

Q: Use the Quadratic Formula to solve the equation .
- or (Incorrect. You may have correctly added 7 to both sides to get , and then substituted , and into the Quadratic Formula: But you mistakenly thought that was equal to .)
- or (Incorrect. You probably substituted 4 for b and for c in the Quadratic Formula. The first step is to add 7 to both sides of the equation: Now substitute , and into)
- or (Incorrect. You probably substituted for c in the Quadratic Formula. The first step is to add 7 to both sides of the equation: Now substitute , and into)
- or (Correct. Add 7 to both sides of the equation: Substitute , and into the Quadratic Formula:)
A: or

Q: Use the discriminant to determine how many and what kind of solutions the quadratic equation has.
- one real solution (Correct. The equation can be rewritten as: In this equation, , and Therefore, Because the discriminant is 0, there is one real solution.)
- no real or complex solutions (Incorrect. You may have correctly calculated the discriminant , got the value 0, and thought that this meant that there are 0 solutions. A quadratic equation always has either real or complex solutions. The solutions are given by the formula:)
- two real solutions (Incorrect. You may have miscalculated the discriminant , thinking that , but you need to rewrite the equation as . In this equation, , and)
- two complex (nonreal) solutions (Incorrect. You may have miscalculated the discriminant . If the discriminant were negative, there would be two complex solutions. You need to rewrite the equation as . In this equation: , and)
A: one real solution

Q: A basketball is thrown with an initial upward velocity of 25 feet per second from a height of 8 feet above the ground. The equation models the height in feet t seconds after it is thrown. After the ball passes its maximum height, it comes down and then goes into the hoop at a height of 10 feet above the ground. About how long after it was thrown does it go into the hoop?
- 1.56 seconds (Incorrect. You solved the equation , when you should have solved the equation .)
- 1.48 seconds (Correct. You need to solve the equation: Rewrite in standard form: Use the Quadratic Formula: Simplify: The smaller solution () is when the ball is going up. The larger solution, seconds, is the answer.)
- 0.08 seconds (Incorrect. You may have correctly set up and solved the equation . However, this is the time when the ball is 10 feet off the ground on its way up. You want to find out when it is coming down.)
- 1.83 seconds (Incorrect. You solved the equation , when you should have solved the equation .)
A: 1.48 seconds