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GRE Subject Test: Math : Algebra

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Example Questions

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Example Question #291 : Gre Subject Test: Math

Simplify $\frac{2i}{9i+3}$

Possible Answers:

$\frac{-38+2i}{13}$

$\frac{49i+27}{52}$

$\frac{-18-6i}{-90}$

$\frac{23i-6}{72}$

$\frac{3+i}{15}$

Correct answer:

$\frac{3+i}{15}$

Explanation:

$\frac{2i}{9i+3}$

In problems like this, you are expected to simplify by removing i from the denominator. To do this, multiply the numerator and denominator by the conjugate of the denominator (switch the sign between the two terms from either a plus to a minus or vice versa) over itself. The conjugate over itself equals 1 and does not change the value of the expression (any number multiplied by 1 is still that number). Multiplying by the conjugate is the only way to eliminate i since there will be no middle term when we foil.

$\frac{2i}{9i+3}\cdot \frac{9i-3}{9i-3}$

$\frac{2i(9i-3)}{(9i+3)(9i-3)}$

$\frac{18i^{2}-6i}{81i^{2}-9}$

Simplify i squared to be -1 and then combine like terms

$\frac{18(-1)-6i}{81(-1)-9}$

$\frac{-18-6i}{-81-9}$

$\frac{-18-6i}{-90}$

Since each term divides by a greatest common factor of -6 reduce all of the coefficients. It would also be equivalent to divide by 6 to reduce all of the terms.

$\frac{3+i}{15}$

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Example Question #11 : Complex Conjugates

Simplify $\frac{-19i-8}{-5i-2}$

Possible Answers:

$\frac{-30i+26}{13}$

$\frac{-111+2i}{-29}$

$\frac{47+78i}{21}$

$\frac{10i-3}{29}$

$\frac{-78i+49}{5}$

Correct answer:

$\frac{-111+2i}{-29}$

Explanation:

$\frac{-19i-8}{-5i-2}$

$\frac{-19i-8}{-5i-2}\cdot \frac{-5i+2}{-5i+2}$

$\frac{(-19i-8)(-5i+2)}{(-5i-2)(-5i+2)}$

$\frac{95i^{2}+40i-38i-16}{25i^{2}-4}$

Simplify i squared to be -1 and then combine like terms

$\frac{95(-1)+40i-38i-16}{25(-1)-4}$

$\frac{-95+40i-38i-16}{-25-4}$

$\frac{-111+2i}{-29}$

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Example Question #111 : Classifying Algebraic Functions

Simplify $\frac{i-2}{3-i}$

Possible Answers:

$\frac{i-7}{10}$

$\frac{7-i}{10}$

$\frac{10i-2}{7}$

$\frac{5-2i}{10}$

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

Correct answer:

$\frac{i-7}{10}$

Explanation:

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

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

$\frac{(i-2)(3+i)}{(3-i)(3+i)}$

$\frac{3i+i^{2}-6-2i}{9-i^{2}}$

Simplify i squared to be -1 and combine like terms

$\frac{3i+(-1)-6-2i}{9-(-1)}$

$\frac{3i-1-6-2i}{9+1}$

$\frac{i-7}{10}$

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Example Question #51 : Imaginary Numbers & Complex Functions

Simplify $\frac{14i-9}{9+i}$

Possible Answers:

$\frac{54i-18}{25}$

$\frac{-44i+63}{86}$

$\frac{23i-11}{65}$

$\frac{14i+111}{42}$

$\frac{135i-67}{82}$

Correct answer:

$\frac{135i-67}{82}$

Explanation:

$\frac{14i-9}{9+i}$

$\frac{14i-9}{9+i}\cdot \frac{9-i}{9-i}$

$\frac{(14i-9)(9-i)}{(9+i)(9-i)}$

$\frac{126i-81-14i^{2}+9i}{81-i^{2}}$

Simplify i squared to be -1 and combine like terms

$\frac{126i-81-14(-1)+9i}{81-(-1)}$

$\frac{126i-81+14+9i}{81+1}$

$\frac{135i-67}{82}$

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Example Question #111 : Algebra

Simplify $\frac{13i+13}{13i-9}$

Possible Answers:

$\frac{-13i-169}{-250}$

$\frac{-26+143i}{-125}$

$\frac{-13+169i}{250}$

$\frac{26-143i}{250}$

$\frac{26i-143}{125}$

Correct answer:

$\frac{-26+143i}{-125}$

Explanation:

$\frac{13i+13}{13i-9}$

$\frac{13i+13}{13i-9}\cdot \frac{13i+9}{13i+9}$

$\frac{(13i+13)(13i+9)}{(13i-9)(13i+9)}$

$\frac{(169i^{2}+169i+117i+117)}{(169i^{2}-81)}$

Simplify i squared to be -1 and combine like terms

$\frac{(169(-1)+169i+117i+117)}{(169)(-1)-81}$

$\frac{(-169+169i+117i+117)}{(-169-81)}$

$\frac{-52+286i}{-250}$

Each term divides by 2 so make sure to reduce all of the terms

$\frac{-26+143i}{-125}$

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Example Question #111 : Classifying Algebraic Functions

Simplify $\frac{-12i-17}{6i+11}$

Possible Answers:

$\frac{19i+259}{-157}$

$\frac{33i-105}{16}$

$\frac{-72i+13}{99}$

$\frac{-20i-187}{34}$

$\frac{24i-15}{81}$

Correct answer:

$\frac{19i+259}{-157}$

Explanation:

$\frac{-12i-17}{6i+11}$

$\frac{-12i-17}{6i+11}\cdot \frac{6i-11}{6i-11}$

$\frac{(-12i-17)(6i-11)}{(6i+11)(6i-11)}$

$\frac{-72i^{2}+121i-102i+187}{36i^{2}-121}$

Simplify i squared to be -1 and combine like terms

$\frac{-72(-1)+121i-102i+187}{36(-1)-121}$

$\frac{72+121i-102i+187}{-36-121}$

$\frac{19i+259}{-157}$

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Example Question #51 : Imaginary Numbers & Complex Functions

Simplify $\frac{3i+7}{4i-4}$

Possible Answers:

$\frac{4i+9}{20}$

$\frac{6i+7}{18}$

$\frac{11i-2}{6}$

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

Undefined

Correct answer:

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

Explanation:

$\frac{3i+7}{4i-4}$

$\frac{3i+7}{4i-4}\cdot \frac{4i+4}{4i+4}$

$\frac{(3i+7)(4i+4)}{(4i-4)(4i+4)}$

$\frac{12i^{2}+12i+28i+28}{16i^{2}-16}$

Simplify i squared to be -1 and combine like terms

$\frac{12(-1)+12i+28i+28}{16(-1)-16}$

$\frac{-12+12i+28i+28}{-16-16}$

$\frac{40i+16}{-32}$

Since every term divides by 8 make sure to reduce all the terms by that greatest common factor

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

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Example Question #112 : Classifying Algebraic Functions

Simplify

$\frac{5+2i}{-3-2i}$

Possible Answers:

$\frac{16+4i}{3}$

$\frac{i+2}{10}$

6+i

$\frac{-19+4i}{13}$

$\frac{7i+1}{5}$

Correct answer:

$\frac{-19+4i}{13}$

Explanation:

$\frac{5+2i}{-3-2i}$

$\frac{5+2i}{-3-2i}\cdot \frac{-3+2i}{-3+2i}$

$\frac{(5+2i)(-3+2i)}{(-3-2i)(-3+2i)}$

$\frac{-15-6i+10i+4i^{2}}{9-4i^{2}}$

Simplify i squared to be -1 and then combine like terms

$\frac{-15-6i+10i+4(-1)}{9-4(-1)}$

$\frac{-15+4i-4}{9+4}$

$\frac{-19+4i}{13}$

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Example Question #3 : Graphing Functions With Complex Numbers

Write the complex number $1-\sqrt{3}i$ in polar form, where polar form expresses the result in terms of a distance from the origin on the complex plane and an angle from the positive -axis, $\theta$ , measured in radians.

Possible Answers:

$\newline r=2 \newline \newline \theta = \frac{5 \pi}{6}$

$\newline r=\sqrt{3} \newline \newline \theta =2 \pi$

$\newline r=\frac{1}{\sqrt{3}} \newline \newline \theta = \frac{3 \pi}{5}$

$\newline r=2 \newline \newline \theta = \sqrt{3} \pi$

$\newline r=1 \newline \newline \theta = \frac{\sqrt{3} \pi}{6}$

Correct answer:

$\newline r=2 \newline \newline \theta = \frac{5 \pi}{6}$

Explanation:

To see what the polar form of the number is, it helps to draw it on a graph, where the horizontal axis is the imaginary part and the vertical axis the real part. This is called the complex plane.

Vecii

To find the angle $\theta$ , we can find its supplementary angle $\Psi$ and subtract it from $\pi$ radians, so $\theta = \pi - \Psi$ .

Using trigonometric ratios, $\tan \Psi = \frac{1}{\sqrt{3}}$ and $\arctan({\frac{1}{\sqrt{3}}})= \Psi = \frac{\pi}{6}$ .

Then $\theta = \frac{5\pi}{6}$ .

To find the distance , we need to find the distance from the origin to the point $(-\sqrt{3}, 1)$ . Using the Pythagorean Theorem to find the hypotenuse , $r = \sqrt{ \sqrt{3}^2 + 1^2 }$ or r=2 .

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Example Question #1 : Evaluating Logarithms

Solve: log_4 256=x

Possible Answers:

x=10

x=-4

x=4

x=6

Correct answer:

x=4

Explanation:

Step 1: Re-write the log equation as an exponential equation. To do this, take the base of the log function and raise it to the number on the right side of the equal sign. This new exponent is equal to the number to the right of the log base.

4^x=256

Step 2: Re-write the right hand side as a power of 4..

256=4^4

Step 3: Re-write the equation

4^x=4^4

Step 4: We have the same base, so we can equal the exponents..

x=4

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GRE Subject Test: Math : Algebra

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

All GRE Subject Test: Math Resources

Example Questions

Example Question #291 : Gre Subject Test: Math

Example Question #11 : Complex Conjugates

Example Question #111 : Classifying Algebraic Functions

Example Question #51 : Imaginary Numbers & Complex Functions

Example Question #111 : Algebra

Example Question #111 : Classifying Algebraic Functions

Example Question #51 : Imaginary Numbers & Complex Functions

Example Question #112 : Classifying Algebraic Functions

Example Question #3 : Graphing Functions With Complex Numbers

Example Question #1 : Evaluating Logarithms

All GRE Subject Test: Math Resources

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