High School Math : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for High School Math

varsity tutors app store varsity tutors android store

Example Questions

Example Question #8 : Using Expressions With Complex Numbers

Solve the following complex number equation:

\(\displaystyle 2n^2=\frac{-27}{8}\)

Possible Answers:

\(\displaystyle n=\frac{-3i\sqrt{3}}{4}\)

\(\displaystyle n=\frac{3i\sqrt{3}}{4}\)

\(\displaystyle n=\frac{3i\sqrt{3}}{2}\)

\(\displaystyle n=\frac{3i\sqrt{3}}{2}, \frac{-3i\sqrt{3}}{2}\)

\(\displaystyle n=\frac{3i\sqrt{3}}{4}, \frac{-3i\sqrt{3}}{4}\)

Correct answer:

\(\displaystyle n=\frac{3i\sqrt{3}}{4}, \frac{-3i\sqrt{3}}{4}\)

Explanation:

Begin by simplifying the equation. Divide both sides by \(\displaystyle 2\) and take the square root.

\(\displaystyle 2n^2=\frac{-27}{8}\)

\(\displaystyle n^2=\frac{-27}{16}\)

\(\displaystyle n=\frac{\sqrt{-27}}{\sqrt{16}}\)

\(\displaystyle n=\frac{3i\sqrt{3}}{4}, \frac{-3i\sqrt{3}}{4}\)

Example Question #9 : Using Expressions With Complex Numbers

Solve the following complex number equation:

\(\displaystyle 4x^2+75=0\)

Possible Answers:

\(\displaystyle x=\frac{5i\sqrt{3}}{3}, -\frac{5i\sqrt{3}}{3}\)

\(\displaystyle x=\frac{5i\sqrt{3}}{3}\)

\(\displaystyle x=\frac{5i\sqrt{3}}{2}\)

\(\displaystyle x=\frac{5i\sqrt{3}}{2}, -\frac{5i\sqrt{3}}{2}\)

\(\displaystyle x= -\frac{5i\sqrt{3}}{2}\)

Correct answer:

\(\displaystyle x=\frac{5i\sqrt{3}}{2}, -\frac{5i\sqrt{3}}{2}\)

Explanation:

Begin by simplifying the equation. Divide both sides by \(\displaystyle 4\) and take the square root.

\(\displaystyle 4x^2+75=0\)

\(\displaystyle 4x^2=-75\)

\(\displaystyle x^2=\frac{-75}{4}\)

\(\displaystyle x=\frac{\sqrt{-75}}{\sqrt{4}}\)

\(\displaystyle x=\frac{5i\sqrt{3}}{2}, -\frac{5i\sqrt{3}}{2}\)

Example Question #10 : Using Expressions With Complex Numbers

Solvethe following complex number expression for \(\displaystyle n\):

\(\displaystyle 2n^2+18=0\)

Possible Answers:

\(\displaystyle n=3i, -3i\)

\(\displaystyle n=2i, -2i\)

\(\displaystyle n=4i, -4i\)

\(\displaystyle n=6i, -6i\)

\(\displaystyle n=5i, -5i\)

Correct answer:

\(\displaystyle n=3i, -3i\)

Explanation:

Solve the equation using complex numbers:

\(\displaystyle 2n^2+18=0\)

\(\displaystyle 2n^2=-18\)

\(\displaystyle n^2=-9\)

\(\displaystyle n=\sqrt{-9}\)

\(\displaystyle n=3i, -3i\)

Example Question #41 : Algebra Ii

Simplify the following complex number expression:

\(\displaystyle (\sqrt{-6})(\sqrt{-4})(\sqrt{-3})\)

Possible Answers:

\(\displaystyle -6i\sqrt{2}\)

\(\displaystyle -6i\sqrt{3}\)

\(\displaystyle -4i\sqrt{2}\)

\(\displaystyle -6i\sqrt{5}\)

\(\displaystyle -4i\sqrt{3}\)

Correct answer:

\(\displaystyle -6i\sqrt{2}\)

Explanation:

Begin by simplifying the radicals using complex numbers:

\(\displaystyle (\sqrt{-6})(\sqrt{-4})(\sqrt{-3})\)

\(\displaystyle =(i\sqrt{6})(2i)(i\sqrt{3})\) 

Multiply the factors:

\(\displaystyle =(2i^2\sqrt{6})(i\sqrt{3})\)

\(\displaystyle =(2i^3\sqrt{18})\)

Simplify. Remember that \(\displaystyle i^3\) is equivalent to \(\displaystyle -i\).

\(\displaystyle =-2i\sqrt{18}\)

\(\displaystyle =-6i\sqrt{2}\)

Example Question #12 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

\(\displaystyle (3i^3)(2i)^2\)

Possible Answers:

\(\displaystyle 11i\)

\(\displaystyle 9i\)

\(\displaystyle 8i\)

\(\displaystyle 12i\)

\(\displaystyle 10i\)

Correct answer:

\(\displaystyle 12i\)

Explanation:

Begin by completing the square:

\(\displaystyle (3i^3)(2i)^2\)

\(\displaystyle =(3i^3)(4i^2)\) 

Multiply the factors:

\(\displaystyle =(12i^5)\) 

Simplify. Remember that \(\displaystyle i^5\) is equivalent to \(\displaystyle i\).

\(\displaystyle =12i\)

Example Question #13 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

\(\displaystyle (-2i)^4(-4i^3)\)

Possible Answers:

\(\displaystyle 12i\)

\(\displaystyle 36i\)

\(\displaystyle 64i\)

\(\displaystyle 72i\)

\(\displaystyle 80i\)

Correct answer:

\(\displaystyle 64i\)

Explanation:

Begin by completing the square:

\(\displaystyle (-2i)^4(-4i^3)\)

\(\displaystyle =(16i^4)(-4i^3)\) 

Now, multiply the factors:

\(\displaystyle =(16i^4)(-4i^3)\)

\(\displaystyle =(-64i^7)\) 

Simplify. Remember that \(\displaystyle i^7\) is equivalent to \(\displaystyle -i\).

\(\displaystyle =64i\)

 

Example Question #14 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

\(\displaystyle (7+2i)(5-3i)\)

Possible Answers:

\(\displaystyle 44-14i\)

\(\displaystyle 43-13i\)

\(\displaystyle 41-11i\)

\(\displaystyle 40-10i\)

\(\displaystyle 42-12i\)

Correct answer:

\(\displaystyle 41-11i\)

Explanation:

Use the FOIL (First, Outer, Inner, Last) to multiply the complex numbers:

\(\displaystyle (7+2i)(5-3i)\)

\(\displaystyle =35-21i+10i-6i^2\)

Combine like terms and simplify. Remember that \(\displaystyle i^2\) is equivalent to \(\displaystyle -1\).

\(\displaystyle =35-11i-6i^2\)

\(\displaystyle =35-11i+6\)

\(\displaystyle =41-11i\)

Example Question #15 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

\(\displaystyle \frac{4+3i}{1-2i}\)

Possible Answers:

\(\displaystyle \frac{-4+11i}{5}\)

\(\displaystyle \frac{-3+11i}{5}\)

\(\displaystyle \frac{-2+11i}{3}\)

\(\displaystyle \frac{-2+11i}{5}\)

\(\displaystyle \frac{-2+11i}{4}\)

Correct answer:

\(\displaystyle \frac{-2+11i}{5}\)

Explanation:

Begin by multiplying the numerator and denominator by the conjugate of the denominator:

\(\displaystyle =\frac{4+3i}{1-2i}\) \(\displaystyle \cdot \frac{1+2i}{1+2i}\)

\(\displaystyle =\frac{4+8i+3i+6i^2}{1-4i^2}\)

Combine like terms:

\(\displaystyle =\frac{4+11i+6i^2}{1-4i^2}\)

Simplify. Remember that \(\displaystyle i^2\) is equivalent to \(\displaystyle -1\).

\(\displaystyle =\frac{4+11i-6}{1+4}\)

\(\displaystyle =\frac{-2+11i}{5}\)

Example Question #31 : Mathematical Relationships And Basic Graphs

Simplify the following complex number expression:

\(\displaystyle \frac{2-2i}{2+2i}\)

Possible Answers:

\(\displaystyle i\)

\(\displaystyle -i\)

\(\displaystyle i^2\)

\(\displaystyle -i^2\)

\(\displaystyle i^3\)

Correct answer:

\(\displaystyle -i\)

Explanation:

Begin by multiplying the numerator and denominator by the conjugate of the denominator:

\(\displaystyle =\frac{2-2i}{2+2i}\)\(\displaystyle \cdot \frac{2-2i}{2-2i}\)

\(\displaystyle =\frac{4-4i-4i+4i^2}{4-4i^2}\)

Combine like terms:

\(\displaystyle =\frac{4-8i+4i^2}{4-4i^2}\)

Simplify. Remember that \(\displaystyle i^2\) is equivalent to \(\displaystyle -1\).

\(\displaystyle =\frac{4-8i-4}{4+4}\)

\(\displaystyle =\frac{-8i}{8}\)

\(\displaystyle =-i\)

Example Question #17 : Using Expressions With Complex Numbers

Simplify the following expression: 

\(\displaystyle (5 + 3i) + (2 - 6i)\)

Possible Answers:

\(\displaystyle 7 - 3i\)

\(\displaystyle 7 + 9i\)

\(\displaystyle 7 + 3i\)

\(\displaystyle 3 - 3i\)

\(\displaystyle 7 - 9i\)

Correct answer:

\(\displaystyle 7 - 3i\)

Explanation:

Simplifying expressions with complex numbers uses exactly the same process as simplifying expressings with one variable. In this case, \(\displaystyle i\) behaves similarly to any other variable. Thus, we have: 

\(\displaystyle (5 + 3i) + (2 - 6i) = (5 +2) + (3 - 6)i = 7 - 3i\)

Learning Tools by Varsity Tutors