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GRE Subject Test: Math : Imaginary Numbers & Complex Functions

Study concepts, example questions & explanations for GRE Subject Test: Math

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All GRE Subject Test: Math Resources

2 Diagnostic Tests 148 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #15 : Operations On Complex Numbers

$\sqrt{5}\cdot \sqrt{-42}$

Possible Answers:

$-2\sqrt{3}$

$i\sqrt{210}$

$5i\sqrt{14}$

$i\sqrt{47}$

Correct answer:

$i\sqrt{210}$

Explanation:

$\sqrt{5}\cdot \sqrt{-42}$

Take out i (the square root of -1) from the radical and then multiply.

$\sqrt{5}\cdot i\sqrt{42}$

$i\sqrt{210}$

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Example Question #16 : Operations On Complex Numbers

$\sqrt{-45}\cdot \sqrt{72}$

Possible Answers:

$6i\sqrt{10}$

$-\sqrt{10}$

$i\sqrt{10}$

$4i\sqrt{10}$

$18i\sqrt{10}$

Correct answer:

$18i\sqrt{10}$

Explanation:

$\sqrt{-45}\cdot \sqrt{72}$

Take out i (the square root of -1) and then simplify before multiplying.

$i\sqrt{45}\cdot \sqrt{72}$

$i\sqrt{9\cdot 5}\cdot \sqrt{36\cdot 2}$

$3i\sqrt{5}\cdot 6\sqrt{2}$

$18i\sqrt{10}$

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Example Question #17 : Operations On Complex Numbers

$-\sqrt{-16}\cdot \sqrt{-40}$

Possible Answers:

$8\sqrt{5}$

$8\sqrt{10}$

$-4i\sqrt{10}$

$i\sqrt{10}$

$4\sqrt{5}$

Correct answer:

$8\sqrt{10}$

Explanation:

$-\sqrt{-16}\cdot \sqrt{-40}$

Take out i (the square root of -1) from both radicals and then multiply. You are not allowed to first multiply the radicals and then simplify because the roots are negative.

$-i\sqrt{16}\cdot i\sqrt{40}$

$-4i\cdot i\sqrt{4\cdot 10}$

$-4i\cdot 2i\sqrt{10}$

$-8i^{2}\sqrt{10}$

$-8(-1)\sqrt{10}$

$8\sqrt{10}$

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Example Question #20 : Operations On Complex Numbers

$-\sqrt{-20}\cdot -\sqrt{32}$

Possible Answers:

$-8i\sqrt{10}$

$-4\sqrt{10}$

$8\sqrt{10}$

$8i\sqrt{10}$

$-8\sqrt{10}$

Correct answer:

$8i\sqrt{10}$

Explanation:

$-\sqrt{-20}\cdot -\sqrt{32}$

Take out i (the square root of -1) from the radical, simplify, and then multiply.

$-i\sqrt{20}\cdot -\sqrt{32}$

$-i\sqrt{5\cdot 4}\cdot -\sqrt{16\cdot 2}$

$-2i\sqrt{5}\cdot -4\sqrt{2}$

$8i\sqrt{10}$

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Example Question #281 : Gre Subject Test: Math

$\sqrt{-70}\cdot -\sqrt{-50}$

Possible Answers:

$10\sqrt{30}$

$10\sqrt{35}$

$5\sqrt{10}$

$-5\sqrt{5}$

$5\sqrt{20}$

Correct answer:

$10\sqrt{35}$

Explanation:

$\sqrt{-70}\cdot -\sqrt{-50}$

Take out i (the square root of -1) from both radicals, simplify, and then multiply. You are not allowed to first multiply the radicals and then simplify because the roots are negative.

$i\sqrt{70}\cdot -i\sqrt{50}$

$i\sqrt{70}\cdot -i\sqrt{25\cdot 2}$

$i\sqrt{70}\cdot -5i\sqrt{2}$

$-5i^{2}\sqrt{140}$

$-5(-1)\sqrt{140}$

$5\sqrt{140}$

$5\sqrt{4\cdot 35}$

$10\sqrt{35}$

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Example Question #31 : Imaginary Numbers & Complex Functions

$\sqrt{-225}\cdot \sqrt{-289}$

Possible Answers:

25i

255i

$-15\sqrt{17}$

$15i\sqrt{17}$

Correct answer:

Explanation:

$\sqrt{-225}\cdot \sqrt{-289}$

Take out i (the square root of -1) from both radicals, simplify, and then multiply. You are not allowed to first multiply the radicals and then simplify because the roots are negative.

$i\sqrt{225}\cdot i\sqrt{289}$

$15i\cdot 17i$

$255i^{2}$

255(-1)

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Example Question #91 : Algebra

$\sqrt{-15}\cdot \sqrt{-33}$

Possible Answers:

$-3\sqrt{55}$

$-i\sqrt{495}$

$i\sqrt{495}$

$-\sqrt{495}$

$\sqrt{495}$

Correct answer:

$-3\sqrt{55}$

Explanation:

$\sqrt{-15}\cdot \sqrt{-33}$

Take out i (the square root of -1) from both radicals and then multiply. You are not allowed to first multiply the radicals and then simplify because the roots are negative.

$i\sqrt{15}\cdot i\sqrt{33}$

$i^{2}\sqrt{495}$

$(-1)\sqrt{495}$

It might be difficult to tell that this simplifies but you should always check if the perfect squares (4, 9, 16, etc) divide into this number. 9 does divide into 495 so this simplifies further.

$-\sqrt{9\cdot 55}$

$-3\sqrt{55}$

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Example Question #281 : Gre Subject Test: Math

$-\sqrt{121}\cdot \sqrt{-12}$

Possible Answers:

$-44\sqrt{3}$

$-22i\sqrt{3}$

$44i\sqrt{3}$

$-44i\sqrt{3}$

$11i\sqrt{12}$

Correct answer:

$-22i\sqrt{3}$

Explanation:

$-\sqrt{121}\cdot \sqrt{-12}$

Take out i (the square root of -1) from the radical, simplify each radical, then multiply.

$-\sqrt{121}\cdot i\sqrt{12}$

$-11\cdot i\sqrt{4\cdot 3}$

$-11\cdot 2i\sqrt{3}$

$-22i\sqrt{3}$

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Example Question #1 : Imaginary Numbers

Evaluate $\small \frac{-6+2i}{10-3i}$

Possible Answers:

You cannot divide by complex numbers

$\small \small \frac{-66+2i}{91-60i}$

$\small \frac{-54+38i}{109}$

$\small \small \frac{-54+38i}{91-60i}$

$\small \frac{-66+2i}{109}$

Correct answer:

$\small \frac{-66+2i}{109}$

Explanation:

To divide by a complex number, we must transform the expression by multiplying it by the complex conjugate of the denominator over itself. In the problem, $\small 10-3i$ is our denominator, so we will multiply the expression by $\small \frac{10+3i}{10+3i}$ to obtain:

$\small \frac{-6+2i}{10-3i}\frac{10+3i}{10+3i}=\frac{-60+20i-18i+6i^2}{100-30i+30i-9i^2}$ .

We can then combine like terms and rewrite all $\small i^2$ terms as $\small -1$ . Therefore, the expression becomes:

$\small \frac{-60+2i+6i^2}{100-9i^2}=\frac{-66+2i}{109}$

Our final answer is therefore $\small \frac{-66+2i}{109}$

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Example Question #1 : Complex Conjugates

Simplify:

$\frac{5-2i}{4-i}$

Possible Answers:

$\frac{22-3i}{17}$

6+3i

18-9i

$\frac{5i+6}{6}$

Correct answer:

$\frac{22-3i}{17}$

Explanation:

To get rid of the fraction, multiply the numerator and denominator by the conjugate of the denominator.

$\frac{5-2i}{4-i}\left(\frac{4+i}{4+i}\right)$

Now, multiply and simplify.

$\frac{5-2i}{4-i}\left(\frac{4+i}{4+i}\right)=\frac{20+5i-8i-2i^2}{16-i^2}$

Remember that i^2=-1

$\frac{20+5i-8i-2i^2}{16-i^2}=\frac{20-3i-2(-1)}{16-(-1)}=\frac{22-3i}{17}$

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GRE Subject Test: Math : Imaginary Numbers & Complex Functions

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #15 : Operations On Complex Numbers

Example Question #16 : Operations On Complex Numbers

Example Question #17 : Operations On Complex Numbers

Example Question #20 : Operations On Complex Numbers

Example Question #281 : Gre Subject Test: Math

Example Question #31 : Imaginary Numbers & Complex Functions

Example Question #91 : Algebra

Example Question #281 : Gre Subject Test: Math

Example Question #1 : Imaginary Numbers

Example Question #1 : Complex Conjugates

All GRE Subject Test: Math Resources

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