GMAT Math : Understanding exponents

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Example Questions

Example Question #11 : Understanding Exponents

Divide:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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Example Question #12 : Understanding Exponents

Solve for $\displaystyle N$:

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

Possible Answers:

The equation has no solution.

N = -2 $\displaystyle N = -2$

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

$\displaystyle -2N + 6 = 3$

$\displaystyle -2N + 6-6 = 3 -6$

$\displaystyle -2N = -3$

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

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

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Example Question #1111 : Gmat Quantitative Reasoning

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

Possible Answers:

$\log 9$ $\displaystyle \log 9$

$\displaystyle 2$

$2 \log 5$ $\displaystyle 2 \log 5$

$\displaystyle 1$

$\displaystyle 0$

Correct answer:

$\displaystyle 1$

Explanation:

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

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

$= \log 40 - \log 4$ $\displaystyle = \log 40 - \log 4$

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

$= \log 10 = 1$ $\displaystyle = \log 10 = 1$

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Example Question #1112 : Gmat Quantitative Reasoning

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

Possible Answers:

$\log 4 - \log 9$ $\displaystyle \log 4 - \log 9$

$\log 2 - \log 3$ $\displaystyle \log 2 - \log 3$

$\displaystyle 0$

$\log 3 - \log 2$ $\displaystyle \log 3 - \log 2$

$\log 9 - \log 4$ $\displaystyle \log 9 - \log 4$

Correct answer:

$\log 9 - \log 4$ $\displaystyle \log 9 - \log 4$

Explanation:

$4 \log 3 - 2 \log 6$ $\displaystyle 4 \log 3 - 2 \log 6$

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

$= \log 81 - \log 36$ $\displaystyle = \log 81 - \log 36$

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

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

$= \log 9 - \log 4$ $\displaystyle = \log 9 - \log 4$

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Example Question #1121 : Gmat Quantitative Reasoning

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

Therefore,

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

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

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

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Example Question #11 : Exponents

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

Possible Answers:

$\log 600$ $\displaystyle \log 600$

$\log 36$ $\displaystyle \log 36$

$\log 12$ $\displaystyle \log 12$

$\log 8$ $\displaystyle \log 8$

$\log 64$ $\displaystyle \log 64$

Correct answer:

$\log 600$ $\displaystyle \log 600$

Explanation:

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

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Example Question #1122 : Gmat Quantitative Reasoning

Express the result in scientific notation:

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

Possible Answers:

$1.25 \times 10^ {11}$ $\displaystyle 1.25 \times 10^ {11}$

$1.25 \times 10^ {20}$ $\displaystyle 1.25 \times 10^ {20}$

$125 \times 10^{9 }$ $\displaystyle 125 \times 10^{9 }$

$1.25 \times 10^ {10}$ $\displaystyle 1.25 \times 10^ {10}$

$125 \times 10^{18 }$ $\displaystyle 125 \times 10^{18 }$

Correct answer:

$1.25 \times 10^ {20}$ $\displaystyle 1.25 \times 10^ {20}$

Explanation:

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

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

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

$= 125 \times 10^{18 }$ $\displaystyle = 125 \times 10^{18 }$

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

$125 \times 10^{18 }$ $\displaystyle 125 \times 10^{18 }$

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

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

$= 1.25 \times 10^ {20}$ $\displaystyle = 1.25 \times 10^ {20}$

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Example Question #13 : Understanding Exponents

Express the quotient in scientific notation:

$\left ( 2.4 \times 10 ^{12}\right ) \div \left (6.4 \times 10^{4} \right )$ $\displaystyle \left ( 2.4 \times 10 ^{12}\right ) \div \left (6.4 \times 10^{4} \right )$

Possible Answers:

$\frac{15}{4}\times 10 ^ {7 }$ $\displaystyle \frac{15}{4}\times 10 ^ {7 }$

$3.75\times 10 ^ {7 }$ $\displaystyle 3.75\times 10 ^ {7 }$

$0.375 \times 10 ^{8}$ $\displaystyle 0.375 \times 10 ^{8}$

$\frac{3}{8} \times 10 ^{8}$ $\displaystyle \frac{3}{8} \times 10 ^{8}$

$3.75\times 10 ^ {9 }$ $\displaystyle 3.75\times 10 ^ {9 }$

Correct answer:

$3.75\times 10 ^ {7 }$ $\displaystyle 3.75\times 10 ^ {7 }$

Explanation:

$\left ( 2.4 \times 10 ^{12}\right ) \div \left (6.4 \times 10^{4} \right )$ $\displaystyle \left ( 2.4 \times 10 ^{12}\right ) \div \left (6.4 \times 10^{4} \right )$

$= \frac{ 2.4 \times 10 ^{12} }{6.4 \times 10^{4}}$ $\displaystyle = \frac{ 2.4 \times 10 ^{12} }{6.4 \times 10^{4}}$

$= \frac{ 2.4 }{6.4 } \times \frac{ 10 ^{12} }{ 10^{4}}$ $\displaystyle = \frac{ 2.4 }{6.4 } \times \frac{ 10 ^{12} }{ 10^{4}}$

$=0.375 \times 10 ^{12-4}$ $\displaystyle =0.375 \times 10 ^{12-4}$

$=0.375 \times 10 ^{8}$ $\displaystyle =0.375 \times 10 ^{8}$

Since 0.375 is not in the range $\left [1,10 \right )$ $\displaystyle \left [1,10 \right )$, we adjust the answer as follows:

$0.375 \times 10 ^{8}$ $\displaystyle 0.375 \times 10 ^{8}$

$= 3.75\times 10 ^ {-1} \times 10 ^{8}$ $\displaystyle = 3.75\times 10 ^ {-1} \times 10 ^{8}$

$= 3.75\times 10 ^ {-1+8 }$ $\displaystyle = 3.75\times 10 ^ {-1+8 }$

$= 3.75\times 10 ^ {7 }$ $\displaystyle = 3.75\times 10 ^ {7 }$

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Example Question #1123 : Gmat Quantitative Reasoning

Give the result in scientific notation:

$\left ( 4 \times 10 ^{-7}\right ) ^{3}$ $\displaystyle \left ( 4 \times 10 ^{-7}\right ) ^{3}$

Possible Answers:

$6.4 \times 10 ^{-10}$ $\displaystyle 6.4 \times 10 ^{-10}$

$64 \times 10 ^{-4}$ $\displaystyle 64 \times 10 ^{-4}$

$6.4 \times 10 ^{-22}$ $\displaystyle 6.4 \times 10 ^{-22}$

$64 \times 10 ^{-21}$ $\displaystyle 64 \times 10 ^{-21}$

$6.4 \times 10 ^{-20 }$ $\displaystyle 6.4 \times 10 ^{-20 }$

Correct answer:

$6.4 \times 10 ^{-20 }$ $\displaystyle 6.4 \times 10 ^{-20 }$

Explanation:

$\left ( 4 \times 10 ^{-7}\right ) ^{3}$ $\displaystyle \left ( 4 \times 10 ^{-7}\right ) ^{3}$

$= 4^{3} \times \left ( 10 ^{-7}\right ) ^{3}$ $\displaystyle = 4^{3} \times \left ( 10 ^{-7}\right ) ^{3}$

$= 64 \times 10 ^{-7\cdot 3}$ $\displaystyle = 64 \times 10 ^{-7\cdot 3}$

$= 64 \times 10 ^{-21}$ $\displaystyle = 64 \times 10 ^{-21}$

Since 64 does not fall in the range [1, 10) $\displaystyle [1, 10)$, we adjust as follows:

$64 \times 10 ^{-21}$ $\displaystyle 64 \times 10 ^{-21}$

$= 6.4\times 10 ^{1} \times 10 ^{-21}$ $\displaystyle = 6.4\times 10 ^{1} \times 10 ^{-21}$

$= 6.4 \times 10 ^{1+ \left (-21 \right )}$ $\displaystyle = 6.4 \times 10 ^{1+ \left (-21 \right )}$

$= 6.4 \times 10 ^{-20 }$ $\displaystyle = 6.4 \times 10 ^{-20 }$

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Example Question #13 : Exponents

Give the sum in scientific notation:

$(8.4 \times 10^{8} ) + (5.7 \times 10 ^{7})$ $\displaystyle (8.4 \times 10^{8} ) + (5.7 \times 10 ^{7})$

Possible Answers:

$1.41 \times 10^{16}$ $\displaystyle 1.41 \times 10^{16}$

$14.1 \times 10^{15}$ $\displaystyle 14.1 \times 10^{15}$

$8.97 \times 10^{8}$ $\displaystyle 8.97 \times 10^{8}$

$89.7 \times 10 ^{7}$ $\displaystyle 89.7 \times 10 ^{7}$

$8.97 \times 10^{16}$ $\displaystyle 8.97 \times 10^{16}$

Correct answer:

$8.97 \times 10^{8}$ $\displaystyle 8.97 \times 10^{8}$

Explanation:

Rewrite the second addend as follows:

$(8.4 \times 10^{8} ) + (5.7 \times 10 ^{7})$ $\displaystyle (8.4 \times 10^{8} ) + (5.7 \times 10 ^{7})$

$=(8.4 \times 10^{8} ) + (0.57 \times 10 ^{1} \times 10 ^{7})$ $\displaystyle =(8.4 \times 10^{8} ) + (0.57 \times 10 ^{1} \times 10 ^{7})$

$= (8.4 \times 10^{8} ) + (0.57 \times 10 ^{1+7} )$ $\displaystyle = (8.4 \times 10^{8} ) + (0.57 \times 10 ^{1+7} )$

$= (8.4 \times 10^{8} ) + (0.57 \times 10 ^{8} )$ $\displaystyle = (8.4 \times 10^{8} ) + (0.57 \times 10 ^{8} )$

$= (8.4 + 0.57 ) \times 10 ^{8}$ $\displaystyle = (8.4 + 0.57 ) \times 10 ^{8}$

$= 8.97 \times 10 ^{8}$ $\displaystyle = 8.97 \times 10 ^{8}$

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GMAT Math : Understanding exponents

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Understanding Exponents

Example Question #12 : Understanding Exponents

Example Question #1111 : Gmat Quantitative Reasoning

Example Question #1112 : Gmat Quantitative Reasoning

Example Question #1121 : Gmat Quantitative Reasoning

Example Question #11 : Exponents

Example Question #1122 : Gmat Quantitative Reasoning

Example Question #13 : Understanding Exponents

Example Question #1123 : Gmat Quantitative Reasoning

Example Question #13 : Exponents

All GMAT Math Resources

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