We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

SAT Math : Squaring / Square Roots / Radicals

Study concepts, example questions & explanations for SAT Math

Create An Account Create Tests & Flashcards

All SAT Math Resources

16 Diagnostic Tests 660 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 3 4 5 6 Next →

Example Question #2 :How To Multiply Complex Numbers

$x^{4}-1$ has 4 roots, including the complex numbers. Take the product of $\left ( 5+2i \right )$ with each of these roots. Take the sum of these 4 results. Which of the following is equal to this sum?

Possible Answers:

The correct answer is not listed.

10i

3+7i

4-10i

Correct answer:

Explanation:

$x^{4}-1=\left ( x^{2} -1\right )\left ( x^{2}+1 \right )=\left ( x+1 \right )\left ( x-1 \right )\left ( x-i \right )\left ( x+i \right )$

This gives us roots of

$1,\ -1,\ i,\ -i$

The product of $\left ( 5+2i \right )$ with each of these gives us:

$(5+2i)\cdot i = 5i + 2i^2 = 5i +2(-1) = 5i-2 = -2 + 5i$

$(5+2i)\cdot (-i) = -5i -2i^2 = -5i -2(-1) = -5i +2 = 2-5i$

$(5+2i)\cdot 1 = 5+2i$

$(5+2i)\cdot (-1) = -5 -2i$

The sum of these 4 is:

[(-2+5i) + (2-5i)] + [(5+2i) + (-5-2i)] =

=[(-2)+2 +5i - 5i] + [5-5 + 2i - 2i] = [0] + [0] = 0

What we notice is that each of the roots has a negative. It thus makes sense that they will all cancel out. Rather than going through all the multiplication, we can instead look at the very beginning setup, which we can simplify using the distributive property:

$1(5+2i) + (-1)\cdot(5+2i)+ i(5+2i)+(-i) \cdot(5+2i) = (5+2i)\cdot[1+(-1)+i+(-i)]$

$= (5+2i)\cdot[0] = 0$

Report an Error

Example Question #3 :How To Multiply Complex Numbers

Simplify:

$(3i)^{6}$

Possible Answers:

729

None of the other responses gives the correct answer.

729 i

-729 i

Correct answer:

Explanation:

Apply the Power of a Product Property:

$(3i)^{6} = 3 ^{6} \cdot i^{6} = 729 \cdot i^{6}$

A power ofcan be found by dividing the exponent by 4 and noting the remainder. 6 divided by 4 is equal to 1, with remainder 2, so

$i^{6} = i^{2} = -1$

Substituting,

$(3i)^{6} = 729 \cdot (-1) = -729$ .

Report an Error

Example Question #4 :How To Multiply Complex Numbers

Multiply $3 + i \sqrt {3 }$ by its complex conjugate.

Possible Answers:

None of the other responses gives the correct answer.

Correct answer:

Explanation:

The complex conjugate of a complex number a+ bi is a - bi . The product of the two is the number

$a^{2}+b^{2}$ .

Therefore, the product of $3 + i \sqrt {3 }$ and its complex conjugate $3 - i \sqrt {3 }$ can be found by setting a = 3 and $b = \sqrt{3}$ in this pattern:

$a^{2}+b^{2} = 3^{2}+ (\sqrt{3})^{2} = 9 + 3 = 12$ ,

the correct response.

Report an Error

Example Question #5 :How To Multiply Complex Numbers

Multiply $11 - i \sqrt{11}$ by its complex conjugate.

Possible Answers:

110

132 i

132

110i

Correct answer:

132

Explanation:

The complex conjugate of a complex number a - bi is a+ bi . The product of the two is the number

$a^{2}+b^{2}$ .

Therefore, the product of $11 - i \sqrt{11}$ and its complex conjugate $11 + i \sqrt{11}$ can be found by setting a = 11 and $b = \sqrt{11}$ in this pattern:

$a^{2}+b^{2} = 11^{2}+ (\sqrt{11})^{2} = 121+11 = 132$ ,

the correct response.

Report an Error

Example Question #6 :How To Multiply Complex Numbers

What is the product of $\frac{3}{2}+ \frac{1}{4} i$ and its complex conjugate?

Possible Answers:

The correct response is not among the other choices.

$-\frac{35}{16} i$

$\frac{35}{16}$

$\frac{35}{16} i$

$-\frac{35}{16}$

Correct answer:

The correct response is not among the other choices.

Explanation:

The complex conjugate of a complex number a+bi is a - bi , so $\frac{3}{2} + \frac{1}{4} i$ has $\frac{3}{2} - \frac{1}{4} i$ as its complex conjugate.

The product of a+bi and a - bi is equal to $a^{2}+ b^{2}$ , so set $a = \frac{3}{2} , b= \frac{1}{4}$ in this expression, and evaluate:

$a^{2}+ b^{2} = \left ( \frac{3}{2} \right )^{2} + \left (\frac{1}{4} \right )^{2} = \frac{9}{4} + \frac{1}{16} = \frac{37}{16}$ .

This is not among the given responses.

Report an Error

Example Question #7 :How To Multiply Complex Numbers

Multiply and simplify:

$\sqrt{-35} \cdot \sqrt{-5}$

Possible Answers:

$-7i \sqrt{5}$

$7i \sqrt{5}$

$-5i \sqrt{7}$

$5i \sqrt{7}$

None of the other choices gives the correct response.

Correct answer:

None of the other choices gives the correct response.

Explanation:

The two factors are both square roots of negative numbers, and are therefore imaginary. Write both in terms ofbefore multiplying:

$\sqrt{-35} = i \sqrt{35}$

$\sqrt{-5} = i \sqrt{5}$

Therefore, using the Product of Radicals rule:

$\sqrt{-35} \cdot \sqrt{-5}$

$= i \sqrt{35} \cdot i \sqrt{5}$

$= i \cdot i \cdot \sqrt{35} \cdot \sqrt{5}$

$= i ^{2}\cdot \sqrt{35 \cdot 5 }$

$=-1 \cdot \sqrt{175 }$

$=-1 \cdot \sqrt{25 }\cdot \sqrt{7}$

$=-1 \cdot 5\cdot \sqrt{7}$

$=-5\sqrt{7}$

Report an Error

Example Question #8 :How To Multiply Complex Numbers

x = 6+ 3i

y= 6 - 3i

Evaluate $x^{3} + 3x^{2} y + 3 xy^{2} + y^{3}$

Possible Answers:

216 i

-216 i

1,728

1,728 -216 i

- 1,728

Correct answer:

1,728

Explanation:

$x^{3} + 3x^{2} y + 3 xy^{2} + y^{3}$ is recognizable as the cube of the binomial x+y . That is,

$x^{3} + 3x^{2} y + 3 xy^{2} + y^{3} = (x+y) ^{3}$

Therefore, setting x = 6+ 3i and y= 6 - 3i and evaluating:

$x^{3} + 3x^{2} y + 3 xy^{2} + y^{3} =[ (6+3i) + (6-3i) ]^{3}$

$= (6+6+3i -3i) ^{3}$

$=12 ^{3}$

= 1,728 .

Report an Error

Example Question #9 :How To Multiply Complex Numbers

x = 7+ 4i

y= 7 - 4i

Evaluate $x^{3} - 3x^{2} y + 3 xy^{2} - y^{3}$

Possible Answers:

None of the other choices gives the correct response.

512i

-512i

343

Correct answer:

-512i

Explanation:

$x^{3} - 3x^{2} y + 3 xy^{2} - y^{3}$ is recognizable as the cube of the binomial x-y . That is,

$x^{3} - 3x^{2} y + 3 xy^{2} - y^{3}= (x-y) ^{3}$

Therefore, setting x = 7+ 4i and y= 7 - 4i and evaluating:

$x^{3} - 3x^{2} y + 3 xy^{2} - y^{3}=[ (7+4i) - (7-4 i) ]^{3}$

$= (7-7 +4i+4i ) ^{3}$

$=(8i)^{3}$

Applying the Power of a Product Rule and the fact that $i^{3} = -i$ :

$(8i)^{3} = 8 ^{3} i ^{3} = 512 (-i) = -512i$ ,

the correct value.

Report an Error

Example Question #10 :How To Multiply Complex Numbers

Raise $6- i\sqrt{7}$ to the power of 3.

Possible Answers:

$342- 115i\sqrt{7}$

$90- 101i \sqrt{7}$

$216 - 7i\sqrt{7}$

$90- 115i\sqrt{7}$

$342- 101i \sqrt{7}$

Correct answer:

$90- 101i \sqrt{7}$

Explanation:

To raise any expression A-B to the third power, use the pattern

$\left (A+B \right )^{3} = A^{3} - 3A^{2} B + 3 AB^{2} - B^{3}$

Setting $A = 6, B = i \sqrt{7}$ :

$(6- i \sqrt{7} )^{3} = 6^{3} - 3\cdot 6^{2} \cdot i \sqrt{7} + 3 \cdot 6 \cdot ( i \sqrt{7})^{2} - ( i \sqrt{7})^{3}$

Taking advantage of the Power of a Product Rule:

$(6- i \sqrt{7} )^{3} = 6^{3} - 3\cdot 6^{2} \cdot i \sqrt{7} + 3 \cdot 6 \cdot i^{2} \cdot ( \sqrt{7})^{2} - i ^{3}\cdot (\sqrt{7})^{3}$

Since $i^{2} = -1$ ,

and

$i^{3} = i^{2} \cdot i = -1 \cdot i = -i$ :

$(6- i \sqrt{7} )^{3}=216 - 3\cdot 36 \cdot i \sqrt{7} + 3 \cdot 6 \cdot 7 \cdot (-1) - (-i) \cdot 7\cdot \sqrt{7}$

$(6- i \sqrt{7} )^{3} =216 - 108i \sqrt{7} -126 + 7i\sqrt{7}$

Collecting real and imaginary terms:

$(6- i \sqrt{7} )^{3} =216 -126 - 108i \sqrt{7} + 7i\sqrt{7}$

$(6- i \sqrt{7} )^{3} =90- 101i \sqrt{7}$

Report an Error

Example Question #201 :Exponents

Raise 6-7i to the power of 3.

Possible Answers:

1,099 - 413 i

None of the other choices gives the correct response.

-666 - 413 i

-666 - 1,099 i

1,099 - 1,099 i

Correct answer:

-666 - 413 i

Explanation:

To raise any expression A-B to the third power, use the pattern

$\left (A+B \right )^{3} = A^{3} - 3A^{2} B + 3 AB^{2} - B^{3}$

Setting A = 6, B = 7i :

$(6-7i)^{3} = 6^{3} - 3\cdot 6^{2} \cdot 7i + 3 \cdot 6 \cdot (7i)^{2} - (7i)^{3}$

Taking advantage of the Power of a Product Rule:

$(6-7i)^{3} = 6^{3} - 3\cdot 6^{2} \cdot 7i + 3 \cdot 6 \cdot 7^{2} \cdot i^{2} - 7^{3} \cdot i^{3}$

Since $i^{2} = -1$ ,

and

$i^{3} = i^{2} \cdot i = -1 \cdot i = -i$ :

$(6-7i)^{3} =216 - 3\cdot 36 \cdot 7i + 3 \cdot 6 \cdot 49 \cdot (-1) - 343 \cdot (-i)$

$(6-7i)^{3} =216 - 756 i -882 + 343 i$

Collecting real and imaginary terms:

$(6-7i)^{3} =216 -882 - 756 i + 343 i$

$(6-7i)^{3} =-666 - 413 i$

Report an Error

← Previous 1 2 3 4 5 6 Next →

View SAT Mathematics Tutors

Evan
Certified Tutor

Drexel University, Bachelor of Science, Business Administration and Management.

View SAT Mathematics Tutors

Harrison
Certified Tutor

University of Central Florida, Bachelor of Science, Multi-/Interdisciplinary Studies, General.

View SAT Mathematics Tutors

Chase
Certified Tutor

Western Governors University, Bachelor of Science, Mathematics.

All SAT Math Resources

16 Diagnostic Tests 660 Practice Tests Question of the Day Flashcards Learn by Concept

SAT Math Tutors in Top Cities:

Atlanta SAT Math Tutors,Austin SAT Math Tutors,Boston SAT Math Tutors,Chicago SAT Math Tutors,Dallas Fort Worth SAT Math Tutors,Denver SAT Math Tutors,Houston SAT Math Tutors,Kansas City SAT Math Tutors,Los Angeles SAT Math Tutors,Miami SAT Math Tutors,New York City SAT Math Tutors,Philadelphia SAT Math Tutors,Phoenix SAT Math Tutors,San Diego SAT Math Tutors,San Francisco-Bay Area SAT Math Tutors,Seattle SAT Math Tutors,St. Louis SAT Math Tutors,Tucson SAT Math Tutors,Washington DC SAT Math Tutors

Popular Courses & Classes

Spanish Courses & Classes in Dallas Fort Worth,GRE Courses & Classes in Miami,MCAT Courses & Classes in New York City,LSAT Courses & Classes in Atlanta,SAT Courses & Classes in Seattle,ISEE Courses & Classes in Houston,SAT Courses & Classes in Washington DC,ISEE Courses & Classes in Washington DC,GMAT Courses & Classes in Atlanta,GRE Courses & Classes in Philadelphia

Popular Test Prep

GMAT Test Prep in Los Angeles,SAT Test Prep in Denver,LSAT Test Prep in New York City,SSAT Test Prep in Washington DC,ISEE Test Prep in New York City,LSAT Test Prep in Houston,LSAT Test Prep in Phoenix,SAT Test Prep in Atlanta,ACT Test Prep in Philadelphia,SSAT Test Prep in Philadelphia

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

*Full legal name

Company name

*Phone number

*Email address

Address

*Address1

Address2

*City

*State

*Zipcode

*Country

Complaint Details

*URL of copyrighted work (where your original material is located)

*Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

我有一个好的信念,垫的使用erial in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

*Signature (your digital signature is legally binding)

高中

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

SAT Math : Squaring / Square Roots / Radicals

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #2 :How To Multiply Complex Numbers

Example Question #3 :How To Multiply Complex Numbers

Example Question #4 :How To Multiply Complex Numbers

Example Question #5 :How To Multiply Complex Numbers

Example Question #6 :How To Multiply Complex Numbers

Example Question #7 :How To Multiply Complex Numbers

Example Question #8 :How To Multiply Complex Numbers

Example Question #9 :How To Multiply Complex Numbers

Example Question #10 :How To Multiply Complex Numbers

Example Question #201 :Exponents

All SAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Find the Best Tutors

Email address:
Your name:
Feedback: