GMAT Math : Exponents

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Example Question #1177 : Gmat Quantitative Reasoning

$4^{t} = \frac{2^{x+3}}{8^{y - 5}}$

Which of the following is a true statement?

Possible Answers:

$t= \frac{1}{2} x - \frac{3}{2} y-6$

t = 2x-6y- 24

t = 2x-6y+ 36

t = 2x-6y+12

$t= \frac{1}{2} x - \frac{3}{2} y+9$

Correct answer:

$t= \frac{1}{2} x - \frac{3}{2} y+9$

Explanation:

$4^{t} = \frac{2^{x+3}}{8^{y - 5}}$

$(2^{2}) ^{t} = \frac{2^{x+3}}{(2^{3}) ^{y - 5}}$

$2^{2t} = \frac{2^{x+3}}{2^{3(y-5)}}$

$2^{2t} = \frac{2^{x+3}}{2^{3y-15}}$

$2^{2t} = 2^{(x+3)- (3y-15)}$

2t=(x+3)- (3y-15)

2t=x+3- 3y+15

2t=x - 3y+18

$\frac{1}{2} \cdot 2t= \frac{1}{2} \cdot ( x - 3y+18 )$

$t= \frac{1}{2} x - \frac{3}{2} y+9$

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Example Question #1178 : Gmat Quantitative Reasoning

$0.25 ^{t} = 2 ^{3y}$

Express in terms of .

Possible Answers:

$t = -\frac{2}{3}y$

$t = -\frac{3}{2}y$

$t = \sqrt{3y}$

t = 3y+2

$t = \sqrt[3]{2y}$

Correct answer:

$t = -\frac{3}{2}y$

Explanation:

$0.25 ^{t} = 2 ^{3y}$

$\left ( \frac{1}{4} \right )^{t} = 2 ^{3y}$

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

$2 ^{-2t} = 2 ^{3y}$

-2t = 3y

$\frac{-2t }{-2 }= \frac{3y}{-2}$

$t = -\frac{3}{2}y$

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Example Question #1179 : Gmat Quantitative Reasoning

$0.04 ^{t} = 25 ^{x+2}$

Express in terms of .

Possible Answers:

$t = \frac{-1}{x+2}$

t = -x+2

$t = \frac{1}{x+2}$

t = x-2

t = -x-2

Correct answer:

t = -x-2

Explanation:

$0.04 ^{t} = 25 ^{x+2}$

$\left ( \frac{1}{25} \right )^{t} = 25 ^{x+2}$

$\left ( 25 ^{-1} \right )^{t} = 25 ^{x+2}$

$25 ^{- t} = 25 ^{x+2}$

-t = x+2

t = -x-2

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Example Question #1180 : Gmat Quantitative Reasoning

The first and third terms of a geometric sequence are and $K ^{4}$ , in that order. Give the eighth term.

(Assume is positive.)

Possible Answers:

$K^{25}$

$K^{11 } \sqrt{K}$

$K^{22}$

$K^{14}$

$K^{10} \sqrt{K}$

Correct answer:

$K^{11 } \sqrt{K}$

Explanation:

Let be the common ratio of the geometric sequence. Then the third term is $r^{2}$ times the first, so

$r^{2} = \frac{K^{4}}{K} = K^{3}$

and

$r = \sqrt{K^{3}} = K^ { \frac{3}{2}}$ .

The eighth term of the sequence is

$a_{8} = a_{1} \cdot r ^{7}$

$= K \cdot (K^{\frac{3}{2}} )^{7}$

$= K \cdot K^{\frac{21}{2}}$

$= K^{\frac{23}{2}}$

$= K^{11 \frac{1}{2}}$

$= K^{11} \cdot K^{ \frac{1}{2}}$

$= K^{11 } \sqrt{K}$

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Example Question #71 : Exponents

Simplify: $\frac{x^6y^{-4}z^0}{x^{-5}y^{10}z^{-6}}$

Possible Answers:

$\frac{y^{14}}{x^{11}z^6}$

xy^6z^6

$\frac{x^{11}z^{6}}{y^{14}}$

$\frac{x^{11}}{y^{14}}$

$\frac{xz^6}{y^{14}}$

Correct answer:

$\frac{x^{11}z^{6}}{y^{14}}$

Explanation:

To solve, we must first simplify the negative exponents by shifting them to the other side of the fraction:

$\frac{x^6y^{-4}z^0}{x^{-5}y^{10}z^{-6}} = \frac{x^6x^5z^0z^6}{y^{10}y^4}$

Then we can simply the multiplied like bases by adding their exponents:

$\frac{x^6x^5z^0z^6}{y^{10}y^4} = \frac{x^{11}z^6}{y^{14}}$

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Example Question #72 : Exponents

$\frac{(2^6 \cdot 2^7)^3}{2^{40}} =$

Possible Answers:

$\frac{1}{16}$

$\frac{1}{2}$

$\frac{1}{4}$

Correct answer:

$\frac{1}{2}$

Explanation:

First we can simplify the numerator's parentheses by adding the like bases' exponents:

$\frac{(2^6 \cdot 2^7)^3}{2^{40}} = \frac{(2^{13})^3}{2^{40}}$

We can then simplify the numerator further by multiplying the base's exponent by the exponent to which it is raised:

$\frac{(2^{13})^3}{2^{40}} = \frac{2^{39}}{2^{40}}$

We can then subtract the denominator's exponent from the numerator's:

$\frac{1}{2^1} = \frac{1}{2}$

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Example Question #73 : Exponents

Simplify the following expression:

$\frac{343y^7}{49y^4}-9y^3$

Possible Answers:

-2y^3

2y^3

285y^3

3y^2

Correct answer:

-2y^3

Explanation:

Simplify the following expression:

$\frac{343y^7}{49y^4}-9y^3$

Let's begin by simplifying the fraction. Recall that when dividing exponents of similar base, we need to subtract the exponents. We can treat the 343 and the 49 just like regular fractions.

$\frac{343y^7}{49y^4}-9y^3=7y^3-9y^3=-2y^3$

Note that then we perform the subtraction step to get our final answer:

-2y^3

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Example Question #71 : Understanding Exponents

is the multiplicative inverse of ; is the additive inverse of . Which of the following is equal to the expression

$(A+B)^{C+D}$

regardless of the values of the variables?

Possible Answers:

A+B

$(A+B)^{C+D}$ must be an undefined quantity

$A + \frac{1}{A}$

Correct answer:

Explanation:

The multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1. Since is the multiplicative inverse of , then

AB = 1 , or $B = \frac{1}{A}$ .

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of ,

C+D= 0

It follows that

$(A+B)^{C+D} = (A+\frac{1}{A})^{0}$

Any nonzero number raised to the power of 0 is equal to 1. Therefore,

$(A+\frac{1}{A})^{0} = 1$ , the correct choice.

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Example Question #71 : Exponents

is the additive inverse of ; is the multiplicative inverse of . Which of the following is equal to the expression

$(AB)^{CD}$

regardless of the values of the variables?

Possible Answers:

$(AB)^{CD}$ is an undefined quantity.

$A ^{2}$

$-A ^{2}$

Correct answer:

$-A ^{2}$

Explanation:

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of ,

A + B = 0 , or B = -A

The multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1. Since is the multiplicative inverse of , then

CD = 1 .

It follows that

$[A \cdot (-A)] ^{1} =( -A ^{2} )^{1} = -A ^{2}$ , the correct response.

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Example Question #76 : Exponents

is the multiplicative inverse of ; is the additive inverse of . Which of the following is equal to the expression

$(AB)^{CD}$

regardless of the values of the variables?

Possible Answers:

$-A ^{2}$

$A ^{2}$

$(AB)^{CD}$ is an undefined quantity.

Correct answer:

Explanation:

The multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1. Since is the multiplicative inverse of , then

AB = 1 .

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of ,

C+D= 0 , or D = -C .

It follows that

$(AB)^{CD} = 1^{C(-C)}= 1^{-C^{2}} = 1$ , the correct response.

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GMAT Math : Exponents

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1177 : Gmat Quantitative Reasoning

Example Question #1178 : Gmat Quantitative Reasoning

Example Question #1179 : Gmat Quantitative Reasoning

Example Question #1180 : Gmat Quantitative Reasoning

Example Question #71 : Exponents

Example Question #72 : Exponents

Example Question #73 : Exponents

Example Question #71 : Understanding Exponents

Example Question #71 : Exponents

Example Question #76 : Exponents

All GMAT Math Resources

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