GMAT Math : Algebra

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Example Questions

Example Question #3 : Understanding Exponents

What are the last two digits, in order, of $6 ^{789}$ ?

Possible Answers:

Correct answer:

Explanation:

Inspect the first few powers of 6; a pattern emerges.

$6^{1} = 6$

$6^{2} = 36$

$6^{3} = 216$

$6^{4} = 1,296$

$6^{5} = 7,776$

$6^{6} = 46,656$

$6^{7} = 279,936$

$6^{8} = 1,679,616$

$6^ { 9 } =10,077,696$

$6^ { 10} =60,466,176$

As you can see, the last two digits repeat in a cycle of 5.

789 divided by 5 yields a remainder of 4; the pattern that becomes apparent in the above list is that if the exponent divided by 5 yields a remainder of 4, then the power ends in the diigts 96.

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Example Question #8 : Understanding Exponents

Which of the following expressions is equal to the expression

$\left ( x^{3}\right ) ^{4} \cdot \left ( x^{3}\right ) ^{5}$ ?

Possible Answers:

$x^{12}$

$x^{27}$

$x^{180}$

$x^{15}$

$x^{60}$

Correct answer:

$x^{27}$

Explanation:

Use the properties of exponents as follows:

$\left ( x^{3}\right ) ^{4} \cdot \left ( x^{3}\right ) ^{5}$

$=x^{3\; \cdot \; 4} \cdot x^{3\; \cdot \; 5}$

$=x^{12} \cdot x^{15}$

$=x^{12+15}$

$=x^{27}$

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Example Question #2 : Exponents

Simplify:

$\left [ \left ( x ^{5} \right )^{4} \right ]^{3}$

Possible Answers:

12x

$x ^{12}$

60x

$60x ^{12}$

$x ^{60}$

Correct answer:

$x ^{60}$

Explanation:

Apply the power of a power principle twice by multiplying exponents:

$\left [ \left ( x ^{5} \right )^{4} \right ]^{3} = \left ( x ^{5 \cdot 4} \right ) ^{3} = \left ( x ^{20} \right ) ^{3} = x ^{20 \cdot 3} = x ^{60 }$

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Example Question #11 : Exponents

Divide:

$\left (56x^{3} +21x ^{2} -63x + 77 \right )\div 7 x ^{2}$

Possible Answers:

$8 x^{2} +3x + \frac{ 11 }{x }-\frac{9}{ x ^{2} }$

$8 x^{2} +3 -\frac{9}{ x ^{2}}+ \frac{ 11 }{x ^{3}}$

$8 x +3 -\frac{9}{ x }+ \frac{ 11 }{x ^{2}}$

$8 x +3 + \frac{ 11 }{x }-\frac{9}{ x ^{2} }$

$8 x^{2} +3x -\frac{9}{ x }+ \frac{ 11 }{x ^{2}}$

Correct answer:

$8 x +3 -\frac{9}{ x }+ \frac{ 11 }{x ^{2}}$

Explanation:

$\left (56x^{3} +21x ^{2} -63x + 77 \right )\div 7 x ^{2}$

$= \frac{ 56x^{3} }{7 x ^{2}} + \frac{ 21x ^{2} }{7 x ^{2}}- \frac{ 63x }{7 x ^{2}}+ \frac{ 77 }{7 x ^{2}}$

$= \frac{ 56 }{7 } x^{3-2} + \frac{ 21 }{7 }- \frac{ 63 }{7}\cdot \frac{1}{ x ^{2-1}}+ \frac{ 77 }{7} \cdot \frac{1}{ x ^{2}}$

$= 8 x +3 -\frac{9}{ x }+ \frac{ 11 }{x ^{2}}$

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Example Question #31 : Algebra

Solve for :

$\left (\frac{1}{25} \right )^{N} \cdot 5^{6} = 125$

Possible Answers:

$N = \frac{3}{2}$

The equation has no solution.

$N =- \frac{1}{4}$

N = -2

$N =- \frac{1}{2}$

Correct answer:

$N = \frac{3}{2}$

Explanation:

$\left (\frac{1}{25} \right )^{N} \cdot 5^{6} = 125$

$\left ( 5 ^{-2} \right )^{N} \cdot 5^{6} = 5^{3}$

$5 ^{-2 \cdot N } \cdot 5^{6} = 5^{3}$

$5 ^{-2 N +6 } = 5^{3}$

-2N + 6 = 3

-2N + 6-6 = 3 -6

-2N = -3

$-2N \div (-2 ) = -3 \div (-2 )$

$N = \frac{3}{2}$

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Example Question #32 : Algebra

Which of the following is equal to $\log 8 + \log 5 - \log 4$ ?

Possible Answers:

$2 \log 5$

$\log 9$

Correct answer:

Explanation:

$\log 8 + \log 5 - \log 4$

$= \log \left ( 8 \cdot 5 \right ) - \log 4$

$= \log 40 - \log 4$

$= \log \left ( 40 \div 4 \right )$

$= \log 10 = 1$

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Example Question #33 : Algebra

Which of the following is equal to $4 \log 3 - 2 \log 6$ ?

Possible Answers:

$\log 3 - \log 2$

$\log 4 - \log 9$

$\log 9 - \log 4$

$\log 2 - \log 3$

Correct answer:

$\log 9 - \log 4$

Explanation:

$4 \log 3 - 2 \log 6$

$= \log\left ( 3^{4} \right ) - \log \left ( 6^{2} \right )$

$= \log 81 - \log 36$

$= \log\frac{ 81}{36}$

$= \log\frac{ 9}{4}$

$= \log 9 - \log 4$

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Example Question #35 : Algebra

Which of the following is true about $\log_{5} 1,000$ ?

Possible Answers:

$5 <\log_{5} 1,000 < 6$

$4 <\log_{5} 1,000 < 5$

$\log_{5} 1,000 = 6$

$\log_{5} 1,000 = 4$

$\log_{5} 1,000 = 5$

Correct answer:

$4 <\log_{5} 1,000 < 5$

Explanation:

$5 ^{4} = 625; 5 ^{5} = 3,125$

Therefore,

$\log_{5} 625 = 4; \log_{5} 3,125 = 5$

$\log_{5} 625 <\log_{5} 1,000 < \log_{5} 3,125$

$4 <\log_{5} 1,000 < 5$

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Example Question #11 : Exponents

Which of the following is equal to $2 + \log 6$ ?

Possible Answers:

$\log 12$

$\log 36$

$\log 600$

$\log 8$

$\log 64$

Correct answer:

$\log 600$

Explanation:

$2 + \log 6 = \log \left ( 10^{2} \right ) + \log 6 = \log 100 + \log 6 = \log\left ( 100 \cdot 6 \right ) = \log 600$

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Example Question #13 : Understanding Exponents

Express the result in scientific notation:

$\left ( 5\times 10^{6}\right )^{3}$

Possible Answers:

$1.25 \times 10^ {10}$

$1.25 \times 10^ {20}$

$125 \times 10^{18 }$

$125 \times 10^{9 }$

$1.25 \times 10^ {11}$

Correct answer:

$1.25 \times 10^ {20}$

Explanation:

$\left ( 5\times 10^{6}\right )^{3}$

$= 5^{3} \times \left ( 10^{6}\right )^{3}$

$= 125 \times 10^{6\cdot 3 }$

$= 125 \times 10^{18 }$

Since 125 is not in the range $\left [1,10 \right )$ , we adjust the answer as follows:

$125 \times 10^{18 }$

$= 1.25 \times 10^ {2}\times 10^{18 }$

$= 1.25 \times 10^ {2+18}$

$= 1.25 \times 10^ {20}$

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #3 : Understanding Exponents

Example Question #8 : Understanding Exponents

Example Question #2 : Exponents

Example Question #11 : Exponents

Example Question #31 : Algebra

Example Question #32 : Algebra

Example Question #33 : Algebra

Example Question #35 : Algebra

Example Question #11 : Exponents

Example Question #13 : Understanding Exponents

All GMAT Math Resources

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