Title: Unit 16 Quiz
Course: Technical Math for Industry

Q: Simplify:
A: (a choice with no text)
If right: Correct. Factor 63 so that you have a perfect square times a number with no square factors: Use to rewrite the expression: Take the square root of :
If wrong: Incorrect. You may have chosen this answer because . However, , using the rule . Because is not equal to 3, .
If wrong: Incorrect. You may have correctly factored as . But while taking the square root of , you mixed up the places of 3 and 7.
If wrong: Incorrect. You recognized that you can rewrite . Therefore, . However, this result is not equal to . You need to use to rewrite the expression as , and then simplify further.

Q: Approximate to the tenths place.
- 7.8 (Incorrect. You may have chosen this approximation because it falls between 7 and 8 and you know that is between and . However, because 52 is closer to 49 than it is to 64, the approximation will be closer to 7 than to 8. Using trial and error, square different approximations to find the best estimate of .)
- 7.3 (Incorrect. This is a reasonable approximation, but it is not the best approximation to the tenths place. Compare to squares of other decimals between 7 and 8.)
- 7.2 (Correct. 52 is between the perfect squares 49 and 64. So is between and . and Because 52 is closer to 49 than it is to 64, is probably about 7.2 or 7.3. and 52 is closer to 51.84 than 53.29, so the answer is 7.2.)
A: 7.2
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 wrong: Incorrect. You forgot to take the absolute values of and . Because these variables each have an odd exponent, the expression or could have a positive or negative value. You need to write and because the square root symbol means take the positive root.
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 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 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.

Q: Simplify:
A: (a choice with no text)
If right: Correct. Factor 108 to find a perfect cube: Factor to find a perfect cube : Rewrite as a product of radicals: Simplify and multiply:
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 wrong: Incorrect. You found three times the square root of the radical expression instead of finding the cube root of the radical expression. Identify cubed numbers and variables. Rewrite as a product of radicals and simplify.
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: Which of the following radical expressions has an absolute value symbol in its simplified form?
A: (a choice with no text)
If wrong: Incorrect. You may have forgotten that if n is odd, , so . The simplified form does not have an absolute value symbol.
If right: Correct. In general, if n is even, . When , this means that , so the simplified form of .
If wrong: Incorrect. The simplified form of this radical is and does not require an absolute value symbol.
If wrong: Incorrect. You may have forgotten that if n is odd, , so . The simplified form does not have an absolute value symbol.

Q: Write as an expression with a rational exponent.
A: (a choice with no text)
If wrong: Incorrect. You incorrectly used the index of the radical as the exponent in the expression. The rule for converting a radical is .
If wrong: Incorrect. You forgot to use a grouping symbol to show that the entire expression under the radical will be raised to the one-sixth power. This answer is equivalent to .
If wrong: Incorrect. You forgot to use a grouping symbol to show that the entire expression under the radical will be raised to the one-sixth power. This answer is equivalent to .
If right: Correct. Rewrite the radical using a rational exponent. The root, 6, determines the denominator of the fraction, so will be the rational exponent. Use a grouping symbol to show that the entire expression under the radical will be raised to the one-sixth power.

Q: Express in radical form.
A: (a choice with no text)
If wrong: Incorrect. You reversed the placement of the index and the exponent. The rule for rewriting rational exponents is:
If wrong: Incorrect. Because is within parentheses, the exponent 3 refers to both terms in the expression, not just to the variable y. So the radicand is .
If right: Correct. The rule for rewriting rational exponents is: The denominator of the fraction is 2, so the radical will be for a square root. Because there are parentheses, the exponent refers to the quantity within the parentheses.
If wrong: Incorrect. Because is within parentheses, the exponent refers to the entire quantity. You only applied the exponent to y.

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 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 right: Correct. Use the rule : Change to radical form:
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. This expression is equivalent to the correct answer, but it is not in simplest form. Factor 49 and rewrite in exponential form before using the laws of exponents to simplify the expression.
If right: Correct. Rewrite the radical using a rational exponent: Use the laws of exponents to simplify: Factor 49 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 used 4 as the numerator and each factor's exponent as a denominator before simplifying. You should have multiplied each factor's exponent by instead.
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 49 in exponential form, and simplify the exponents.

Q: Which one of the following simplifications is incorrect?
A: (a choice with no text)
If right: Correct. Both radicals are fourth roots but you found square roots, so this is an incorrect simplification.
If wrong: Incorrect. This is a correct simplification because:
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 it is not in simplest form. You need to simplify the variable part of the radicand by identifying perfect cubes and pulling them out.
If right: Correct.
If wrong: Incorrect. Instead of dividing 75 by 3, it appears that you subtracted 3 from 75. You may have confused division of coefficients with division with exponents. You then found the cube root of . Instead, rewrite using . Identify common factors in the numerator and denominator and simplify. Identify perfect cubes and pull them out of the radical.
If wrong: Incorrect. You found the square root of some factors in the radicand instead of the cube root. Rewrite using . Identify common factors in the numerator and denominator and simplify. Identify perfect cubes and pull them out of the radical.

Q: Add and simplify:
A: (a choice with no text)
If wrong: Incorrect. You basically simplified correctly, except for taking the square root of . When you take the square root of a perfect square variable you must include the absolute value symbols.
If right: Correct. Simplify the first and last terms using : The first and last terms are like terms and can be combined: So the entire expression equals . Because the indices are not the same, this expression cannot be further simplified.
If wrong: Incorrect. You may not have noticed the index of the cube root, and so incorrectly combined all the radicals, while not simplifying the square roots. Start by simplifying the first and last terms using . Now the first and last terms are like terms that you can combine, just as you would combine .
If wrong: Incorrect. You have found an equivalent expression, but it is not in simplest form. You may have simplified the third radical and then tried to combine all three terms. Start by simplifying the first and last terms using . Now the first and last 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 combine all three terms. The coefficients are not like terms, so a 2 cannot be combined with and . Only combine the first and second terms in this problem.
If wrong: Incorrect. You may have simplified the radicals correctly, but then tried to divide the two terms. 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. This is equivalent to the original expression but is not in simplest form. You did not simplify the variables. Rewrite each 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 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.
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 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 right: Correct. Use the Distributive Property:

Q: Rationalize the denominator and simplify:
A: (a choice with no text)
If wrong: Incorrect. It appears that you did not distribute to both terms in the numerator. The first two steps are: Then simplify radicals.
If wrong: Incorrect. It appears that you did not distribute to both terms in the numerator. The first two steps are: Then simplify radicals.
If right: Correct. Rationalize the denominator by multiplying the expression by:
If wrong: Incorrect. You may have made a mistake when multiplying the numerator by , obtaining and then simplifying. You should have obtained and then simplified.

Q: Rationalize the denominator and simplify:
A: (a choice with no text)
If wrong: Incorrect. You may have thought that the in the top and the bottom would simplify to . That kind of simplification would hold only if were a factor of the denominator. Multiply the numerator and denominator by the conjugate of the denominator: Use FOIL. Simplify radicals.
If right: Correct. Multiply the numerator and denominator by the conjugate of the denominator:
If wrong: Incorrect. You may have just found the conjugate of the denominator. Multiply the numerator and denominator by the conjugate of the denominator: Use FOIL. Simplify radicals.
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: Distribute on top and use FOIL on the bottom. Simplify radicals.

Q: Solve for x:
- 4.8 (Incorrect. You may have correctly squared both sides, but then solved the resulting linear equation incorrectly. Always perform the same operation on both sides. In this case, after squaring, make sure that you add 1 to both sides.)
- 1.5 (Incorrect. You may have doubled 7 while squaring the other side. Always perform the same operation on both sides. In this case, make sure that you square both sides.)
- 0.8 (Incorrect. You may have squared the radical, but forgot to square the other side. Make sure that you square both sides, and then solve the new equation.)
- 5 (Correct.)
A: 5

Q: Solve:
A: (a choice with no text)
If wrong: Incorrect. You probably squared both sides and solved the resulting quadratic equation correctly. However, you have to check the solutions to the quadratic equation back in the original equation to see what is an actual solution and what is an extraneous solution.
If wrong: Incorrect. You probably squared both sides and tried to solve the resulting quadratic equation by factoring. However, the solutions to are not and . Remember that this equation becomes two linear equations: and . Solve these and then check your results back in the original equation. If a solution does not make the equation true, it is an extraneous 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 right: Correct. , so and Check | ? | | extraneous solution Check : | ? | | TRUE

Q: The radius of a cone can be found using the formula , where V stands for the volume of the cone and h stands for the height. If , , and is 3.14, find the volume (V). Round the answer to the nearest hundredths place.
- 1,893.04 cubic units (Incorrect. You may have correctly started with , simplified to , and then multiplied by 25.12. This will not produce on the right side, though. Square both sides before multiplying or dividing to isolate V.)
- 25.12 cubic units (Incorrect. You may have correctly started with , 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.)
- 75.36 cubic units (Correct. , , so)
- 50.24 cubic units (Incorrect. You may have correctly started with , 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.)
A: 75.36 cubic units

Q: Simplify:
A: (a choice with no text)
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.
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 right: Correct. Use the rule and the definition of i.
If wrong: Incorrect. You may have correctly noticed the perfect square 121 as a factor of 363, 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.

Q: How would the real number be written as a complex number?
A: (a choice with no text)
If wrong: Incorrect. You may have chosen this option because it contains and , but it is not equal to . A complex number is any number in the form where a and b are real numbers. The fraction is the real part of this complex number.
If right: Correct. A complex number is any number in the form where a and b are real numbers. In this case is the imaginary part.
If wrong: Incorrect. This expression is another way to write the fraction , but it is not written as a complex number. A complex number is any number in the form where a and b are real numbers. The fraction is the real part of this complex number.
If wrong: Incorrect. You may have chosen this option because it is in the form of a complex number using the 7 and the 2, but it is not equal to . A complex number is any number in the form where a and b are real numbers. The fraction is the real part of this complex number.

Q: Add:
A: (a choice with no text)
If wrong: Incorrect. You made a mistake when you were combining the imaginary parts. Remember that when you combine like terms you add the coefficients.
If wrong: Incorrect. It appears that you ignored the minus sign in front of the . Rearrange the sum to put like terms together, and then combine those like terms, being careful with your signs.
If right: Correct. Rearrange the sum to put like terms together, and then combine those like terms:
If wrong: Incorrect. You may have tried to combine all the terms together, but they are not like terms. Real numbers are like terms and imaginary numbers are like terms. Combine only like terms.

Q: Subtract:
A: (a choice with no text)
If right: Correct. 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 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 8. 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. It appears that you multiplied the coefficients but not the i's. Use the fact that .
If right: Correct. Multiply the coefficients of i together, and remember that .
If wrong: Incorrect. It appears that you added the coefficients. Multiply the coefficients of i together, and remember that .
If wrong: Incorrect. You found a correct product, but it is not in simplest form. Remember that is equal to .

Q: Multiply and simplify:
- 53 (Incorrect. You may have thought that the problem showed a complex number being multiplied by its conjugate, so you added 4 to 49. 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 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.
If right: Correct. Use FOIL and simplify.

Q: Simplify:
A: (a choice with no text)
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 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 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 .
If right: Correct. Treat the division as a fraction, simplify, and then rationalize the denominator.

Q: Solve:
A: (a choice with no text)
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. It appears that you used the wrong order of operations and solved the equation . The correct first step is to add 7 to both sides to get . Take the square root of both sides, isolate x, and simplify the radical.
If wrong: Incorrect. You probably did the first two steps correctly, but when you got to isolating the variable you added when you should have subtracted 7 from both sides. Also, you need to simplify the radical.
If right: Correct. Add 7 to both sides: Use the Square Root Property: Subtract 7 from 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 324; (Incorrect. You would add 324 to to make it . If you start with , add the number to complete the square. The resulting trinomial will be Here .)
- add 18; (Incorrect. is not equal to . If you start with , add the number to complete the square. The resulting trinomial will be Here .)
- add 36; (Incorrect. is not equal to . If you start with , add the number to complete the square. The resulting trinomial will be Here .)
- add 81; (Correct. If you start with , add the number to complete the square. , so add The resulting trinomial is)
A: add 81;

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 and the terms in the proper order. You did not do this, but directly read off the constants starting with 5.
If wrong: Incorrect. It appears that you rewrote the equation so that the right side is 0, and correctly reordered the terms, but misread a coefficient. Do not forget to include any minus signs that may appear.
If wrong: Incorrect. You probably rewrote the equation so that the right side was 0, but then did not reorder the terms. Standard form of a quadratic equation is . 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 (Incorrect. You probably correctly rewrote the equation as and used the Quadratic Formula, , but divided by 2 instead of .)
- or (Incorrect. You probably correctly rewrote the equation as and used the Quadratic Formula, , but made a mistake when finding .)
- or (Correct. First rewrite as . Then use the Quadratic Formula, , with , and :)
- 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 probably substituted for c in the Quadratic Formula. The first step is to add 14 to both sides of the equation: Now substitute , and into)
- or (Incorrect. You may have correctly added 14 to both sides to get , and then substituted , and into the Quadratic Formula: But you mistakenly thought that was equal to .)
- or (Correct. Add 14 to both sides of the equation: Substitute , and into the Quadratic Formula:)
- or (Incorrect. You probably substituted 6 for b and for c in the Quadratic Formula. The first step is to add 14 to both sides of the equation: Now substitute , and into)
A: or

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

Q: A basketball is thrown with an initial upward velocity of 23 feet per second from a height of 7 feet above the ground. The equation models the height in feet t seconds after the basketball 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?
- 1.70 seconds (Incorrect. You solved the equation , when you should have solved the equation .)
- 0.15 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.)
- 1.29 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.)
- 1.44 seconds (Incorrect. You solved the equation , when you should have solved the equation .)
A: 1.29 seconds