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High School Math : Algebra II

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Example Questions

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Example Question #11 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

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

Possible Answers:

$-4i\sqrt{2}$

$-6i\sqrt{5}$

$-6i\sqrt{3}$

$-4i\sqrt{3}$

$-6i\sqrt{2}$

Correct answer:

$-6i\sqrt{2}$

Explanation:

Begin by simplifying the radicals using complex numbers:

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

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

Multiply the factors:

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

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

Simplify. Remember that i^3 is equivalent to .

$=-2i\sqrt{18}$

$=-6i\sqrt{2}$

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Example Question #12 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

(3i^3)(2i)^2

Possible Answers:

11i

10i

12i

Correct answer:

12i

Explanation:

Begin by completing the square:

(3i^3)(2i)^2

=(3i^3)(4i^2)

Multiply the factors:

=(12i^5)

Simplify. Remember that i^5 is equivalent to .

=12i

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Example Question #13 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

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

Possible Answers:

80i

72i

36i

12i

64i

Correct answer:

64i

Explanation:

Begin by completing the square:

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

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

Now, multiply the factors:

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

=(-64i^7)

Simplify. Remember that i^7 is equivalent to .

=64i

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Example Question #14 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

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

Possible Answers:

43-13i

40-10i

42-12i

44-14i

41-11i

Correct answer:

41-11i

Explanation:

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

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

=35-21i+10i-6i^2

Combine like terms and simplify. Remember that i^2 is equivalent to .

=35-11i-6i^2

=35-11i+6

=41-11i

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Example Question #15 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

$\frac{4+3i}{1-2i}$

Possible Answers:

$\frac{-4+11i}{5}$

$\frac{-2+11i}{5}$

$\frac{-2+11i}{3}$

$\frac{-3+11i}{5}$

$\frac{-2+11i}{4}$

Correct answer:

$\frac{-2+11i}{5}$

Explanation:

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

$=\frac{4+3i}{1-2i}$ $\cdot \frac{1+2i}{1+2i}$

$=\frac{4+8i+3i+6i^2}{1-4i^2}$

Combine like terms:

$=\frac{4+11i+6i^2}{1-4i^2}$

Simplify. Remember that i^2 is equivalent to .

$=\frac{4+11i-6}{1+4}$

$=\frac{-2+11i}{5}$

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Example Question #16 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

$\frac{2-2i}{2+2i}$

Possible Answers:

i^3

i^2

-i^2

Correct answer:

Explanation:

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

$=\frac{2-2i}{2+2i}$ $\cdot \frac{2-2i}{2-2i}$

$=\frac{4-4i-4i+4i^2}{4-4i^2}$

Combine like terms:

$=\frac{4-8i+4i^2}{4-4i^2}$

Simplify. Remember that i^2 is equivalent to .

$=\frac{4-8i-4}{4+4}$

$=\frac{-8i}{8}$

=-i

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Example Question #17 : Using Expressions With Complex Numbers

Simplify the following expression:

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

Possible Answers:

7 + 9i

3 - 3i

7 + 3i

7 - 9i

7 - 3i

Correct answer:

7 - 3i

Explanation:

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

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

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Example Question #18 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

(3+8i)(3-8i)

Possible Answers:

Correct answer:

Explanation:

Use the FOIL (First, Outer, Inner, Last) method to expand the expression:

(3+8i)(3-8i)

=9-24i+24i-64i^2

=9-64i^2

Simplify. Remember that i^2 is equivalent to .

=9+64

=73

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Example Question #1 : Understanding Square Roots

Simplify the expression. Find the positive solution only.

$\sqrt{50x^3y^4}$

Possible Answers:

$25x^2y^4\sqrt{2x}$

$5xy^2\sqrt{2x}$

$xy^2\sqrt{50xy^2}$

25xy

$5xy\sqrt{2x}$

Correct answer:

$5xy^2\sqrt{2x}$

Explanation:

When working in square roots, each component can be treated separately.

$\sqrt{50x^3y^4}=\sqrt{50}\sqrt{x^3}\sqrt{y^4}$

Now, we can simplify each term.

$\sqrt{50}=5\sqrt{2}$

$\sqrt{x^3}=x\sqrt{x}$

$\sqrt{y^4}=y^2$

Combine the simplified terms to find the answer. Anything outside of the square root is combined, while anything under the root is combined under the root.

$(5\sqrt{2})(x\sqrt{x})(y^2)=5xy^2\sqrt{2x}$

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Example Question #1 : Understanding Radicals

Convert the radical to exponential notation.

$\small \sqrt[4]{13}$

Possible Answers:

$13^{\frac{1}{4}}$

$\small 13^4$

$\small 169$

$\small \sqrt{13^4}$

Correct answer:

$13^{\frac{1}{4}}$

Explanation:

Remember that any term outside the radical will be in the denominator of the exponent.

$\sqrt[b]{x^a}=x^{\frac{a}{b}}$

Since does not have any roots, we are simply raising it to the one-fourth power.

$\sqrt[4]{13}=13^{\frac{1}{4}}$

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High School Math : Algebra II

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #11 : Using Expressions With Complex Numbers

Example Question #12 : Using Expressions With Complex Numbers

Example Question #13 : Using Expressions With Complex Numbers

Example Question #14 : Using Expressions With Complex Numbers

Example Question #15 : Using Expressions With Complex Numbers

Example Question #16 : Using Expressions With Complex Numbers

Example Question #17 : Using Expressions With Complex Numbers

Example Question #18 : Using Expressions With Complex Numbers

Example Question #1 : Understanding Square Roots

Example Question #1 : Understanding Radicals

All High School Math Resources

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