GMAT Math : Exponents

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Example Question #21 : Exponents

Give the sum in scientific notation:

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

Possible Answers:

$10.48 \times 10^ {8}$

$17.13 \times 10 ^{15}$

$1.713 \times 10 ^{16}$

$1.048 \times 10^ {9}$

$1.048 \times 10^ {16}$

Correct answer:

$1.048 \times 10^ {9}$

Explanation:

Rewrite the second addend as follows:

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

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

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

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

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

$= 10.48 \times 10^ {8}$

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

$10.48 \times 10^ {8}$

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

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

$= 1.048 \times 10^ {9}$

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Example Question #22 : Exponents

Give the quotient in scientific notation:

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

Possible Answers:

$0.9375 \times 10^{-4}$

$9.375\times 10^{ -5}$

$9.375\times 10^{ -14}$

$0.9375 \times 10^{-15}$

$9.375\times 10^{ -16}$

Correct answer:

$9.375\times 10^{ -16}$

Explanation:

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

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

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

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

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

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

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

$0.9375 \times 10^{-15}$

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

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

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

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Example Question #23 : Exponents

Give the difference in scientific notation:

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

Possible Answers:

$7\times 10^{15}$

$1 \times 10^{3}$

$7\times 10^{3}$

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

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

Correct answer:

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

Explanation:

Rewrite the second addend as follows:

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

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

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

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

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Example Question #24 : Exponents

Express the product in scientific notation:

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

Possible Answers:

$4 \times 10 ^{16}$

$40 \times 10 ^{15}$

$0.4 \times 10 ^{58}$

$4 \times 10 ^{57}$

$40 \times 10 ^{56}$

Correct answer:

$4 \times 10 ^{16}$

Explanation:

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

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

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

$= 40 \times 10 ^{15 }$

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

$40 \times 10 ^{15 }$

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

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

$= 4 \times 10 ^{16 }$

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Example Question #41 : Algebra

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

Possible Answers:

125

100

Correct answer:

125

Explanation:

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

Since x^2=5 , we have

(x^2)^3=(x^2)(x^2)(x^2)=(5)(5)(5)=125 .

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Example Question #26 : Exponents

Simplify:

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

Possible Answers:

$32x ^{10}$

$4,096 x ^{12}$

$4,096 x ^{10}$

$24x ^{12}$

$32x ^{12}$

Correct answer:

$32x ^{12}$

Explanation:

Use the properties of exponents as follows:

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

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

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

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

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

$=32x^{12}$

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Example Question #27 : Exponents

Solve for :

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

Possible Answers:

N = 6

The equation has no solution.

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

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

N= -2

Correct answer:

N= -2

Explanation:

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

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

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

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

4N + 5 = -3

4N + 5 -5 = -3 -5

4N = -8

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

N= -2

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Example Question #28 : Exponents

$M = \log A$

$N = \log B$

$P = \log C$

Which of the following is equal to $\log \frac{100 A^{2} B}{C ^{3}}$ ?

Possible Answers:

$2 + M^{2} + N - P^{3}$

$100 + M^{2} + N - P^{3}$

100 + 2 M + N - 3P

2 + 2 M + N - 3P

$\frac{ 4 M N }{ 3P}$

Correct answer:

2 + 2 M + N - 3P

Explanation:

We'll need to remember a few logarithmic properties to answer this question:

$log(A\times B)=log(A)+log(B)$

$log(\frac{A}{B})=log(A)-log(B)$

$log(A^x)=x\cdot log(A)$

Now we can use these same rules to rewrite the log in question:

$\log \frac{100 A^{2} B}{C ^{3}}$

$= \log \left ( 100 A^{2} B \right )- \log \left ( C ^{3} \right )$

$= \log 100 + \log \left ( A^{2} \right ) + \log B - \log \left ( C ^{3} \right )$

$= \log 100 + 2 \log A + \log B - 3 \log C$

=2 + 2 M + N - 3P

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Example Question #29 : Exponents

Simplify the following expression:

$\frac{(x^{2+3})^{4}}{x^{12}}$ .

Possible Answers:

$x^{\frac{5}{3}}$

$x^{-3}$

$x^{32}$

$x^{2}$

$x^{8}$

Correct answer:

$x^{8}$

Explanation:

We start by simplifying the expression on the top. Let's add the exponents inside the parentheses and then multiply by the exponent outside of the parentheses.

Then we substract the denominator exponent form the numerator exponent:

$\frac{(x^{2+3})^{4}}{x^{12}}=\frac{(x^{5})^{4}}{x^{12}}=\frac{x^{5*4}}{x^{12}}=\frac{x^{20}}{x^{12}}=x^{20-12}=x^{8}$

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Example Question #30 : Exponents

Simplify the following:

$a^{6}b^{-4}c^{-2}\cdot \frac{a^7b^{11}c^{13}}{a^{13}b^{7}c^{11}}}$ .

Possible Answers:

$a^{26}b^{14}c^{26}$

$a^{14}b^{8}c^{4}$

$a^{-13}b^{13}c^{13}$

abc

Correct answer:

Explanation:

First, we need to add the exponents of the elements with the same base that are multiplied, and subtract the exponents of same-base elements that are divided:

$\frac{a^7b^{11}c^{13}}{a^{13}b^{7}c^{11}}}=a^{-6}b^{4}c^{2}$

Then $a^{6}b^{-4}c^{-2}\cdot a^{-6}b^{4}c^{-2}=a^{0}b^{0}c^{0} or (abc)^{0}}$

Any value raised to the power of 0 equals 1 so the final result is 1.

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GMAT Math : Exponents

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #21 : Exponents

Example Question #22 : Exponents

Example Question #23 : Exponents

Example Question #24 : Exponents

Example Question #41 : Algebra

Example Question #26 : Exponents

Example Question #27 : Exponents

Example Question #28 : Exponents

Example Question #29 : Exponents

Example Question #30 : Exponents

All GMAT Math Resources

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