GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #91 : Algebra

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

Possible Answers:

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

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

xy^6z^6 $\displaystyle xy^6z^6$

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

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

Correct answer:

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

Explanation:

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

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

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

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

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Example Question #71 : Exponents

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

Possible Answers:

$\frac{1}{16}$ $\displaystyle \frac{1}{16}$

$\frac{1}{4}$ $\displaystyle \frac{1}{4}$

$\displaystyle 4$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 2$

Correct answer:

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

Explanation:

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

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

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

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

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

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

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Example Question #71 : Understanding Exponents

Simplify the following expression:

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

Possible Answers:

3y^2 $\displaystyle 3y^2$

2y^3 $\displaystyle 2y^3$

-2y^3 $\displaystyle -2y^3$

285y^3 $\displaystyle 285y^3$

Correct answer:

-2y^3 $\displaystyle -2y^3$

Explanation:

Simplify the following expression:

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

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.

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

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

-2y^3 $\displaystyle -2y^3$

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Example Question #1181 : Gmat Quantitative Reasoning

$\displaystyle A$ is the multiplicative inverse of $\displaystyle B$; $\displaystyle C$ is the additive inverse of $\displaystyle D$. Which of the following is equal to the expression

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

regardless of the values of the variables?

Possible Answers:

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

$\displaystyle 1$

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

A+B $\displaystyle A+B$

$\displaystyle 0$

Correct answer:

$\displaystyle 1$

Explanation:

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

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

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since $\displaystyle C$ is the additive inverse of $\displaystyle D$ ,

C+D= 0 $\displaystyle C+D= 0$

It follows that

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

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

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

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Example Question #71 : Exponents

$\displaystyle A$ is the additive inverse of $\displaystyle B$; $\displaystyle C$ is the multiplicative inverse of $\displaystyle D$. Which of the following is equal to the expression

$(AB)^{CD}$ $\displaystyle (AB)^{CD}$

regardless of the values of the variables?

Possible Answers:

$\displaystyle 0$

$\displaystyle 1$

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

$-A ^{2}$ $\displaystyle -A ^{2}$

$A ^{2}$ $\displaystyle A ^{2}$

Correct answer:

$-A ^{2}$ $\displaystyle -A ^{2}$

Explanation:

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since $\displaystyle A$ is the additive inverse of $\displaystyle B$,

$\displaystyle A + B = 0$, or B = -A $\displaystyle B = -A$

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

CD = 1 $\displaystyle CD = 1$.

It follows that

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

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Example Question #1183 : Gmat Quantitative Reasoning

$\displaystyle A$ is the multiplicative inverse of $\displaystyle B$; $\displaystyle C$ is the additive inverse of $\displaystyle D$. Which of the following is equal to the expression

$(AB)^{CD}$ $\displaystyle (AB)^{CD}$

regardless of the values of the variables?

Possible Answers:

$A ^{2}$ $\displaystyle A ^{2}$

$\displaystyle 0$

$\displaystyle 1$

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

$-A ^{2}$ $\displaystyle -A ^{2}$

Correct answer:

$\displaystyle 1$

Explanation:

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

AB = 1 $\displaystyle AB = 1$.

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since $\displaystyle C$ is the additive inverse of $\displaystyle D$ ,

C+D= 0 $\displaystyle C+D= 0$, or D = -C $\displaystyle D = -C$.

It follows that

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

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Example Question #81 : Understanding Exponents

$\displaystyle A$ is the additive inverse of the multiplicative inverse of $\displaystyle B$; $\displaystyle C$ is the additive inverse of the multiplicative inverse of $\displaystyle D$. Which of the following is equal to the expression

$(AB)^{CD}$ $\displaystyle (AB)^{CD}$

regardless of the values of the variables?

Possible Answers:

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

$\displaystyle 1$

$\displaystyle 2$

$\displaystyle -1$

$\displaystyle 0$

Correct answer:

$\displaystyle -1$

Explanation:

The additive inverse of a number is the number which, when added to that number, yields sum 0; the multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1.

Let $\displaystyle X$ be the multiplicative inverse of $\displaystyle B$. Then

BX= 1 $\displaystyle BX= 1$, or, equivalently, $X = \frac{1}{B}$ $\displaystyle X = \frac{1}{B}$.

$\displaystyle A$ is the additive inverse of this number, so

$\displaystyle A + X = 0$

$A + \frac{1}{B}= 0$ $\displaystyle A + \frac{1}{B}= 0$

$A = - \frac{1}{B}$ $\displaystyle A = - \frac{1}{B}$

$A \cdot B = - \frac{1}{B} \cdot B$ $\displaystyle A \cdot B = - \frac{1}{B} \cdot B$

AB = -1 $\displaystyle AB = -1$

By similar reasoning, CD = -1 $\displaystyle CD = -1$, and

$(AB)^{CD} = \left ( -1\right ) ^{-1} = \frac{1}{-1} = -1$ $\displaystyle (AB)^{CD} = \left ( -1\right ) ^{-1} = \frac{1}{-1} = -1$

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Example Question #81 : Exponents

$FG= 2^{100}$ $\displaystyle FG= 2^{100}$

$HJ = 2^{10}$ $\displaystyle HJ = 2^{10}$

$\frac{F}{J} = 4^{40}$ $\displaystyle \frac{F}{J} = 4^{40}$

Which of the following is equal to $\frac{H}{G}$ $\displaystyle \frac{H}{G}$ ?

Possible Answers:

$\frac{1}{2^{170}}$ $\displaystyle \frac{1}{2^{170}}$

$\frac{1}{2^{50}}$ $\displaystyle \frac{1}{2^{50}}$

$\frac{1}{2^{10}}$ $\displaystyle \frac{1}{2^{10}}$

$\frac{1}{2^{130}}$ $\displaystyle \frac{1}{2^{130}}$

$\frac{1}{2^{70}}$ $\displaystyle \frac{1}{2^{70}}$

Correct answer:

$\frac{1}{2^{10}}$ $\displaystyle \frac{1}{2^{10}}$

Explanation:

Divide:

$\frac{HJ}{FG}=\frac{ 2^{10}}{2^{100}} = 2^{10-100} = 2^{-90}$ $\displaystyle \frac{HJ}{FG}=\frac{ 2^{10}}{2^{100}} = 2^{10-100} = 2^{-90}$

$\frac{HJ}{FG}= \frac{H}{G} \cdot \frac{J}{F} = \frac{H}{G} \div \frac{F} {J}$ $\displaystyle \frac{HJ}{FG}= \frac{H}{G} \cdot \frac{J}{F} = \frac{H}{G} \div \frac{F} {J}$

$\frac{F}{J} = 4^{40} = (2 ^{2})^{40} = 2 ^{2 \cdot 40 } = 2 ^{80}$ $\displaystyle \frac{F}{J} = 4^{40} = (2 ^{2})^{40} = 2 ^{2 \cdot 40 } = 2 ^{80}$

Substitute:

$\frac{H}{G} \div \frac{F} {J} = \frac{HJ}{FG}$ $\displaystyle \frac{H}{G} \div \frac{F} {J} = \frac{HJ}{FG}$

$\frac{H}{G} \div 2 ^{80}= 2^{-90}$ $\displaystyle \frac{H}{G} \div 2 ^{80}= 2^{-90}$

$\frac{H}{G} \div 2 ^{80}\cdot 2 ^{80} = 2^{-90} \cdot 2 ^{80}$ $\displaystyle \frac{H}{G} \div 2 ^{80}\cdot 2 ^{80} = 2^{-90} \cdot 2 ^{80}$

$\frac{H}{G} = 2^{-90+ 80} = 2^{-10} = \frac{1}{2^{10}}$ $\displaystyle \frac{H}{G} = 2^{-90+ 80} = 2^{-10} = \frac{1}{2^{10}}$

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Example Question #82 : Exponents

$\displaystyle A$ is the additive inverse of $\displaystyle B$; $\displaystyle C$ is the multiplicative inverse of $\displaystyle D$. Which of the following is equal to the expression

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

regardless of the values of the variables?

Possible Answers:

$A - \frac{1}{A}$ $\displaystyle A - \frac{1}{A}$

$\displaystyle 1$

$\displaystyle 0$

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

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

Correct answer:

$\displaystyle 0$

Explanation:

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since $\displaystyle A$ is the additive inverse of $\displaystyle B$,

$\displaystyle A + B = 0$

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

CD = 1 $\displaystyle CD = 1$, or $D = \frac{1}{C}$ $\displaystyle D = \frac{1}{C}$.

It follows that

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

0 raised to any nonzero power is equal to 0, and $C + \frac{1}{C}$ $\displaystyle C + \frac{1}{C}$ must be nonzero, so

$0^{C+ \frac{1}{C}} = 0$ $\displaystyle 0^{C+ \frac{1}{C}} = 0$, the correct response.

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Example Question #81 : Understanding Exponents

$\displaystyle A$ is the additive inverse of $\displaystyle B$; $\displaystyle C$ is the additive inverse of $\displaystyle D$. Which of the following is equal to the expression

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

regardless of the values of the variables?

Possible Answers:

$\displaystyle 0$

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

$\displaystyle 1$

$A - \frac{1}{A}$ $\displaystyle A - \frac{1}{A}$

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

Correct answer:

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

Explanation:

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since $\displaystyle A$ is the additive inverse of $\displaystyle B$ and $\displaystyle C$ is the additive inverse of $\displaystyle D$,

$\displaystyle A + B = C+D = 0$

and

$(A+B)^{C+D} = 0^{0}$ $\displaystyle (A+B)^{C+D} = 0^{0}$,

which is an undefined expression.

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GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #91 : Algebra

Example Question #71 : Exponents

Example Question #71 : Understanding Exponents

Example Question #1181 : Gmat Quantitative Reasoning

Example Question #71 : Exponents

Example Question #1183 : Gmat Quantitative Reasoning

Example Question #81 : Understanding Exponents

Example Question #81 : Exponents

Example Question #82 : Exponents

Example Question #81 : Understanding Exponents

All GMAT Math Resources

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