GMAT Math : Understanding exponents

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Example Questions

Example Question #61 : Understanding Exponents

Simplify:

$\frac{(4x^{7}y^{-4}z^{0})^{3}}{(2x^{-3}y^{10}z^1)^4}$

Possible Answers:

$\frac{4x^{33}}{y^{52}z^{4}}}$

Unable to simplify

$\frac{64x^{21}y^{-12}}{16x^{-12}y^{40}z^4}$

$\frac{4x^9}{y^{13}z^2}$

$\frac{64x^{10}y^{-1}z^3}{16xy^{14}z^5}$

Correct answer:

$\frac{4x^{33}}{y^{52}z^{4}}}$

Explanation:

The first thing we must do is distribute the exponents outside of the parentheses across each expression (remembering of course that exponents set to another exponent are multiplied).

$\frac{(4x^{7}y^{-4}z^{0})^{3}}{(2x^{-3}y^{10}z^1)^4} = \frac{4^3(x^7)^3(y^{-4})^3(z^0)^3}{2^4(x^{-3})^4(y^{10})^4(z^1)^4} =$ $\frac{64x^{21}y^{-12}}{16x^{-12}y^{40}z^4}$

The last step is to follow the rules of exponent addition/subtraction:

$a^b \cdot a^c = a^{b+c}$ and $a^b / a^c = a^{b-c}$

Therefore:

$\frac{64x^{21}y^{-12}}{16x^{-12}y^{40}z^4} =$ $\frac{4x^{33}}{y^{52}z^{4}}}$

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Example Question #62 : Exponents

Which of the following is true if $5^{x}= 3,899$ ?

Possible Answers:

8 < x < 9

7 < x < 8

The equation has no solution.

5 < x < 6

6< x < 7

Correct answer:

5 < x < 6

Explanation:

To answer this question, note that

$5^{5}= 3,125$

and

$5^{6}= 15,625$ .

Therefore, since

3,125< 3,899<15,625

it follows that

$5^{5} < 5^{x}< 5^{6}$

and

5 < x < 6 .

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Example Question #62 : Understanding Exponents

Solve for :

$4^{x}= 512$

Possible Answers:

$x= \frac{9}{2}$

x = 4

The equation has no solution.

$x= \frac{1}{4}$

$x= \frac{2}{9}$

Correct answer:

$x= \frac{9}{2}$

Explanation:

Rewrite both sides as powers of 2 using the exponent rules as follows:

$4^{x}= 512$

$(2^{2} )^{x}= 512$

$2 ^{2\cdot x} = 2^{9}$

Since powers of the same base are equal, set the exponents equal to each other:

2x = 9

$x= \frac{9}{2}$

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Example Question #63 : Understanding Exponents

Solve for :

$9^{t} = 3^{t - 4}$

Possible Answers:

t = 4

t = 2

The equation has no solution.

t = -4

t= -2

Correct answer:

t = -4

Explanation:

Rewrite both sides as powers of 3 using the exponent rules as follows:

$9^{t} = 3^{t - 4}$

$(3^{2})^{t} = 3^{t - 4}$

$3^{2\cdot t}= 3^{t - 4}$

Since powers of the same base are equal, set the exponents equal to each other:

2t = t-4

2t - t= t-4 - t

t = -4

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Example Question #64 : Understanding Exponents

Solve for :

$2^{x+1} \cdot 4 ^{x} = 8^{3x}$

Possible Answers:

x = 1

x= 6

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

The equation has no solution.

$x = \frac{1}{6}$

Correct answer:

$x = \frac{1}{6}$

Explanation:

Rewrite all expressions as powers of 2, and use the exponent rules as follows:

$2^{x+1} \cdot \left ( 2^{2}\right ) ^{x} = \left ( 2^{3}\right ) ^{3x}$

$2^{x+1} \cdot 2^{2 \cdot x} = 2^{3 \cdot 3x }$

$2^{x+1} \cdot 2^{2x} = 2^{9x }$

$2^{x+1 + 2x} = 2^{9x }$

$2^{3x+1 } = 2^{9x }$

Since powers of the same base are equal, set the exponents equal to each other:

3x+1=9x

3x+1 - 3x=9x - 3x

1 = 6x

$x = \frac{1}{6}$

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Example Question #1172 : Gmat Quantitative Reasoning

Solve for :

$2 ^{x} + 2 ^{x-1} = 24$

Possible Answers:

$x = \frac{9}{2}$

x= 4

The equation has no solution.

x= 3

$x= \frac{7}{2}$

Correct answer:

x= 4

Explanation:

Rewrite the first expression to get:

$2 ^{x} + 2 ^{x-1} = 24$

$2 \cdot 2 ^{x-1} + 2 ^{x-1} = 24$

$2 \cdot 2 ^{x-1} + 1 \cdot 2 ^{x-1} = 24$

$3 \cdot 2 ^{x-1} = 24$

$\frac{3 \cdot 2 ^{x-1} }{3}= \frac{24}{3}$

$2 ^{x-1} = 8$

$2 ^{x-1} = 2^{3}$

x-1 = 3

x=4

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Example Question #65 : Understanding Exponents

Solve for :

$2^{3t} = 8^{t - 4}$

Possible Answers:

t = 0

t = 3

t = -4

t = 1

The equation has no solution.

Correct answer:

The equation has no solution.

Explanation:

Rewrite both sides as powers of 2 using the exponent rules as follows:

$2^{3t} = 8^{t - 4}$

$2^{3t} = (2^{3}) ^{t - 4}$

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

Since powers of the same base are equal, set the exponents equal to each other:

3t = 3t-12

3t - 3t= 3t-12 - 3t

0 = -12

This is identically false, so the equation has no solution.

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Example Question #1171 : Problem Solving Questions

Simplify:

$\frac{(x^3y^4z)^2}{x^{\frac{1}{2}}y^9}$

Possible Answers:

$\frac{x^3z^2}{y}$

$\frac{x^{\frac{11}{2}}z^2}{y}$

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

$\frac{x^{\frac{5}{2}}z^2}{y^5}$

$\frac{x^5z^2}{y^{10}}$

Correct answer:

$\frac{x^{\frac{11}{2}}z^2}{y}$

Explanation:

The first step is two distribute the squared exponent across the numerator:

$\frac{(x^3y^4z)^2}{x^{\frac{1}{2}}y^9} = \frac{x^6y^8z^2}{x^{\frac{1}{2}}y^9}$

We can then subtract the denominator's exponents from the numerator's leaving us with the answer:

$\frac{x^6y^8z^2}{x^{\frac{1}{2}}y^9} = \frac{x^{\frac{11}{2}} z^2}{y}$

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Example Question #61 : Exponents

The first two terms of a geometric sequence are and $C ^{3}$ , in that order. Give the tenth term.

(Assume is positive.)

Possible Answers:

$C^{11}$

$C^{19}$

$C^{21}$

$C^{27}$

$C^{30}$

Correct answer:

$C^{19}$

Explanation:

The common ratio of the geometric sequence can be found by dividing the second term by the first:

$\frac{C^{3}}{C} = C^{2}$

The tenth term of the sequence is therefore

$a_{10} = a_{1}r^{10-1}=C \cdot \left ( C^{2}\right ) ^{9} = C \cdot C^{18} = C^{19}$

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Example Question #1176 : Gmat Quantitative Reasoning

$7^{3x+2} = 49 ^{y - 12 }$

Express in terms of .

Possible Answers:

y = 6x - 8

y = 6x+16

$y = \frac{3}{2} x+25$

$y = \frac{3}{2} x-11$

$y = \frac{3}{2} x+13$

Correct answer:

$y = \frac{3}{2} x+13$

Explanation:

$7^{3x+2} = 49 ^{y - 12 }$

$7^{3x+2} = \left (7^{2} \right ) ^{y - 12 }$

$7^{3x+2} = 7^{2 \cdot (y - 12 )}$

$7^{3x+2} = 7^{2 y - 24}$

2y-24 = 3x+2

2y-24 + 24= 3x+2 + 24

2y = 3x+2 6

$\frac{1}{2}\cdot 2y = \frac{1}{2}\cdot (3x+2 6)$

$y = \frac{3}{2} x+13$

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GMAT Math : Understanding exponents

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #61 : Understanding Exponents

Example Question #62 : Exponents

Example Question #62 : Understanding Exponents

Example Question #63 : Understanding Exponents

Example Question #64 : Understanding Exponents

Example Question #1172 : Gmat Quantitative Reasoning

Example Question #65 : Understanding Exponents

Example Question #1171 : Problem Solving Questions

Example Question #61 : Exponents

Example Question #1176 : Gmat Quantitative Reasoning

All GMAT Math Resources

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