GMAT Math : Algebra

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Example Questions

Example Question #41 : Algebra

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:

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

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

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

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

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

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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Example Question #21 : Exponents

Give the sum in scientific notation:

$(9.63 \times 10^{8} ) + \left ( 8.5 \times 10^{7}\right )$ $\displaystyle (9.63 \times 10^{8} ) + \left ( 8.5 \times 10^{7}\right )$

Possible Answers:

$1.048 \times 10^ {16}$ $\displaystyle 1.048 \times 10^ {16}$

$17.13 \times 10 ^{15}$ $\displaystyle 17.13 \times 10 ^{15}$

$10.48 \times 10^ {8}$ $\displaystyle 10.48 \times 10^ {8}$

$1.048 \times 10^ {9}$ $\displaystyle 1.048 \times 10^ {9}$

$1.713 \times 10 ^{16}$ $\displaystyle 1.713 \times 10 ^{16}$

Correct answer:

$1.048 \times 10^ {9}$ $\displaystyle 1.048 \times 10^ {9}$

Explanation:

Rewrite the second addend as follows:

$(9.63 \times 10^{8} ) + \left ( 8.5 \times 10^{7}\right )$ $\displaystyle (9.63 \times 10^{8} ) + \left ( 8.5 \times 10^{7}\right )$

$= (9.63 \times 10^{8} ) + \left ( 0.85 \times 10^ {1} \times 10^{7}\right )$ $\displaystyle = (9.63 \times 10^{8} ) + \left ( 0.85 \times 10^ {1} \times 10^{7}\right )$

$= (9.63 \times 10^{8} ) + \left ( 0.85 \times 10^ {1+7} \right )$ $\displaystyle = (9.63 \times 10^{8} ) + \left ( 0.85 \times 10^ {1+7} \right )$

$= (9.63 \times 10^{8} ) + \left ( 0.85 \times 10^ {8} \right )$ $\displaystyle = (9.63 \times 10^{8} ) + \left ( 0.85 \times 10^ {8} \right )$

$= (9.63 + 0.85 ) \times 10^ {8}$ $\displaystyle = (9.63 + 0.85 ) \times 10^ {8}$

$= 10.48 \times 10^ {8}$ $\displaystyle = 10.48 \times 10^ {8}$

This is not in scientific notation; we adjust as follows:

$10.48 \times 10^ {8}$ $\displaystyle 10.48 \times 10^ {8}$

$= 1.048 \times 10^{1} \times 10^ {8}$ $\displaystyle = 1.048 \times 10^{1} \times 10^ {8}$

$= 1.048 \times 10^ {1+ 8}$ $\displaystyle = 1.048 \times 10^ {1+ 8}$

$= 1.048 \times 10^ {9}$ $\displaystyle = 1.048 \times 10^ {9}$

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Example Question #44 : Algebra

Give the quotient in scientific notation:

$(1.5 \times 10^{-20}) \div (1.6 \times 10^{-5})$ $\displaystyle (1.5 \times 10^{-20}) \div (1.6 \times 10^{-5})$

Possible Answers:

$0.9375 \times 10^{-4}$ $\displaystyle 0.9375 \times 10^{-4}$

$9.375\times 10^{ -5}$ $\displaystyle 9.375\times 10^{ -5}$

$9.375\times 10^{ -16}$ $\displaystyle 9.375\times 10^{ -16}$

$0.9375 \times 10^{-15}$ $\displaystyle 0.9375 \times 10^{-15}$

$9.375\times 10^{ -14}$ $\displaystyle 9.375\times 10^{ -14}$

Correct answer:

$9.375\times 10^{ -16}$ $\displaystyle 9.375\times 10^{ -16}$

Explanation:

$(1.5 \times 10^{-20}) \div (1.6 \times 10^{-5})$ $\displaystyle (1.5 \times 10^{-20}) \div (1.6 \times 10^{-5})$

$=\frac{ 1.5 \times 10^{-20} }{1.6 \times 10^{- 5}}$ $\displaystyle =\frac{ 1.5 \times 10^{-20} }{1.6 \times 10^{- 5}}$

$=\frac{ 1.5 }{1.6 } \times \frac{ 10^{-20} }{ 10^{- 5}}$ $\displaystyle =\frac{ 1.5 }{1.6 } \times \frac{ 10^{-20} }{ 10^{- 5}}$

$=0.9375 \times 10^{-20- (-5)}$ $\displaystyle =0.9375 \times 10^{-20- (-5)}$

$=0.9375 \times 10^{-20+5}$ $\displaystyle =0.9375 \times 10^{-20+5}$

$=0.9375 \times 10^{-15}$ $\displaystyle =0.9375 \times 10^{-15}$

Since 0.9375 does not fall in the range $\left [1,10 \right )$ $\displaystyle \left [1,10 \right )$, we adjust the answer as follows:

$0.9375 \times 10^{-15}$ $\displaystyle 0.9375 \times 10^{-15}$

$= 9.375\times 10^{-1} \times 10^{-15}$ $\displaystyle = 9.375\times 10^{-1} \times 10^{-15}$

$= 9.375\times 10^{-1 + (-15)}$ $\displaystyle = 9.375\times 10^{-1 + (-15)}$

$= 9.375\times 10^{ -16}$ $\displaystyle = 9.375\times 10^{ -16}$

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Example Question #21 : Exponents

Give the difference in scientific notation:

$\left (7 \times 10 ^{9} \right ) - \left (7 \times 10 ^{6} \right )$ $\displaystyle \left (7 \times 10 ^{9} \right ) - \left (7 \times 10 ^{6} \right )$

Possible Answers:

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

$1 \times 10^{3}$ $\displaystyle 1 \times 10^{3}$

$6.93 \times 10 ^{9} \right )$ $\displaystyle 6.93 \times 10 ^{9} \right )$

$6.993 \times 10 ^{9} \right )$ $\displaystyle 6.993 \times 10 ^{9} \right )$

$7\times 10^{3}$ $\displaystyle 7\times 10^{3}$

Correct answer:

$6.993 \times 10 ^{9} \right )$ $\displaystyle 6.993 \times 10 ^{9} \right )$

Explanation:

Rewrite the second addend as follows:

$\left (7 \times 10 ^{9} \right ) - \left (7 \times 10 ^{6} \right )$ $\displaystyle \left (7 \times 10 ^{9} \right ) - \left (7 \times 10 ^{6} \right )$

$= \left (7 \times 10 ^{9} \right ) - \left (0.007 \times 10 ^{9} \right )$ $\displaystyle = \left (7 \times 10 ^{9} \right ) - \left (0.007 \times 10 ^{9} \right )$

$= \left (7 - 0.007 \right ) \times 10 ^{9} \right )$ $\displaystyle = \left (7 - 0.007 \right ) \times 10 ^{9} \right )$

$=6.993 \times 10 ^{9} \right )$ $\displaystyle =6.993 \times 10 ^{9} \right )$

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Example Question #21 : Understanding Exponents

Express the product in scientific notation:

$\left ( 8 \times 10 ^{8}\right ) ( 5 \times 10 ^{7})$ $\displaystyle \left ( 8 \times 10 ^{8}\right ) ( 5 \times 10 ^{7})$

Possible Answers:

$4 \times 10 ^{16}$ $\displaystyle 4 \times 10 ^{16}$

$40 \times 10 ^{56}$ $\displaystyle 40 \times 10 ^{56}$

$4 \times 10 ^{57}$ $\displaystyle 4 \times 10 ^{57}$

$40 \times 10 ^{15}$ $\displaystyle 40 \times 10 ^{15}$

$0.4 \times 10 ^{58}$ $\displaystyle 0.4 \times 10 ^{58}$

Correct answer:

$4 \times 10 ^{16}$ $\displaystyle 4 \times 10 ^{16}$

Explanation:

$\left ( 8 \times 10 ^{8}\right ) ( 5 \times 10 ^{7})$ $\displaystyle \left ( 8 \times 10 ^{8}\right ) ( 5 \times 10 ^{7})$

$= 8 \times 5 \times 10 ^{8} \times 10 ^{7}$ $\displaystyle = 8 \times 5 \times 10 ^{8} \times 10 ^{7}$

$= 40 \times 10 ^{8+7}$ $\displaystyle = 40 \times 10 ^{8+7}$

$= 40 \times 10 ^{15 }$ $\displaystyle = 40 \times 10 ^{15 }$

Since 40 does not fall inside the rangle [1,10) $\displaystyle [1,10)$, we adjust the answer as follows:

$40 \times 10 ^{15 }$ $\displaystyle 40 \times 10 ^{15 }$

$= 4 \times 10^{1} \times 10 ^{15 }$ $\displaystyle = 4 \times 10^{1} \times 10 ^{15 }$

$= 4 \times 10 ^{1+ 15 }$ $\displaystyle = 4 \times 10 ^{1+ 15 }$

$= 4 \times 10 ^{16 }$ $\displaystyle = 4 \times 10 ^{16 }$

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Example Question #42 : Algebra

If x^2=5 $\displaystyle x^2=5$, what does x^6 $\displaystyle x^6$ equal?

Possible Answers:

100 $\displaystyle 100$

$\displaystyle 15$

125 $\displaystyle 125$

$\displaystyle 75$

$\displaystyle 25$

Correct answer:

125 $\displaystyle 125$

Explanation:

We can use the fact that $(x^a)^b=x^{a*b}$ $\displaystyle (x^a)^b=x^{a*b}$ to see that $\displaystyle (x^2)^3=x^6.$

Since x^2=5 $\displaystyle x^2=5$, we have

$\displaystyle (x^2)^3=(x^2)(x^2)(x^2)=(5)(5)(5)=125$.

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Example Question #43 : Algebra

Simplify:

$\left ( 2x^{2} \right )^{3} \cdot \left ( 2x^{3} \right )^{2}$ $\displaystyle \left ( 2x^{2} \right )^{3} \cdot \left ( 2x^{3} \right )^{2}$

Possible Answers:

$32x ^{12}$ $\displaystyle 32x ^{12}$

$32x ^{10}$ $\displaystyle 32x ^{10}$

$4,096 x ^{10}$ $\displaystyle 4,096 x ^{10}$

$24x ^{12}$ $\displaystyle 24x ^{12}$

$4,096 x ^{12}$ $\displaystyle 4,096 x ^{12}$

Correct answer:

$32x ^{12}$ $\displaystyle 32x ^{12}$

Explanation:

Use the properties of exponents as follows:

$\left ( 2x^{2} \right )^{3} \cdot \left ( 2x^{3} \right )^{2}$ $\displaystyle \left ( 2x^{2} \right )^{3} \cdot \left ( 2x^{3} \right )^{2}$

$= 2 ^{3} \cdot \left ( x^{2} \right )^{3} \cdot 2 ^{2} \cdot \left ( x^{3} \right )^{2}$ $\displaystyle = 2 ^{3} \cdot \left ( x^{2} \right )^{3} \cdot 2 ^{2} \cdot \left ( x^{3} \right )^{2}$

$=8 \cdot x^{2 \; \cdot \; 3} \cdot 4 \cdot x^{3 \; \cdot \; 2}$ $\displaystyle =8 \cdot x^{2 \; \cdot \; 3} \cdot 4 \cdot x^{3 \; \cdot \; 2}$

$=32 \cdot x^{6} \cdot x^{6}$ $\displaystyle =32 \cdot x^{6} \cdot x^{6}$

$=32 \cdot x^{6+6}$ $\displaystyle =32 \cdot x^{6+6}$

$=32x^{12}$ $\displaystyle =32x^{12}$

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Example Question #44 : Algebra

Solve for $\displaystyle N$:

$\left (3 ^{4} \right )^{N}\cdot 3^{5} =\frac{1}{27}$ $\displaystyle \left (3 ^{4} \right )^{N}\cdot 3^{5} =\frac{1}{27}$

Possible Answers:

$N = \frac{7}{6}$ $\displaystyle N = \frac{7}{6}$

N = 6 $\displaystyle N = 6$

The equation has no solution.

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

$N = - \frac{3}{20}$ $\displaystyle N = - \frac{3}{20}$

Correct answer:

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

Explanation:

$\left (3 ^{4} \right )^{N}\cdot 3^{5} =\frac{1}{27}$ $\displaystyle \left (3 ^{4} \right )^{N}\cdot 3^{5} =\frac{1}{27}$

$3 ^{4\cdot N} \cdot 3^{5} =3 ^{-3 }$ $\displaystyle 3 ^{4\cdot N} \cdot 3^{5} =3 ^{-3 }$

$3 ^{4\cdot N+5} =3 ^{-3 }$ $\displaystyle 3 ^{4\cdot N+5} =3 ^{-3 }$

The left and right sides of the equation have the same base, so we can equate the exponents and solve:

$\displaystyle 4N + 5 = -3$

$\displaystyle 4N + 5 -5 = -3 -5$

4N = -8 $\displaystyle 4N = -8$

$4N\div 4 = -8 \div 4$ $\displaystyle 4N\div 4 = -8 \div 4$

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

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #41 : Algebra

Example Question #1123 : Gmat Quantitative Reasoning

Example Question #13 : Exponents

Example Question #21 : Exponents

Example Question #44 : Algebra

Example Question #21 : Exponents

Example Question #21 : Understanding Exponents

Example Question #42 : Algebra

Example Question #43 : Algebra

Example Question #44 : Algebra

All GMAT Math Resources

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