GMAT Math : Algebra

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Example Question #1181 : Gmat Quantitative Reasoning

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:

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

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

A+B

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 #1181 : Problem Solving Questions

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 #1183 : Gmat Quantitative Reasoning

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:

$A ^{2}$

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

$-A ^{2}$

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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Example Question #81 : Understanding Exponents

is the additive inverse of the multiplicative inverse of ; is the additive inverse of 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

Correct answer:

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 be the multiplicative inverse of . Then

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

is the additive inverse of this number, so

A + X = 0

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

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

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

AB = -1

By similar reasoning, CD = -1 , and

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

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Example Question #81 : Exponents

$FG= 2^{100}$

$HJ = 2^{10}$

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Divide:

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

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

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

Substitute:

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

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

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

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

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Example Question #82 : Exponents

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

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

regardless of the values of the variables?

Possible Answers:

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

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

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

Correct answer:

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

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

It follows that

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

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

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

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Example Question #84 : Understanding Exponents

is the additive 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:

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

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

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

Correct answer:

$(A+B)^{C+D}$ 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 is the additive inverse of and is the additive inverse of ,

A + B = C+D = 0

and

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

which is an undefined expression.

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Example Question #1 : Solving Inequalities

How many integers $\dpi{100} \small (x)$ can complete this inequality?

$7< 2x-3 <15$

Possible Answers:

$\dpi{100} \small 4$

$\dpi{100} \small 6$

$\dpi{100} \small 5$

$\dpi{100} \small 3$

$\dpi{100} \small 9$

Correct answer:

$\dpi{100} \small 3$

Explanation:

$7< 2x-3 <15$

3 is added to each side to isolate the $\dpi{100} \small x$ term:

$10< 2x <18$

Then each side is divided by 2 to find the range of $\dpi{100} \small x$ :

$5< x <9$

The only integers that are between 5 and 9 are 6, 7, and 8.

The answer is 3 integers.

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Example Question #2 : Solving Inequalities

Solve $5 < 3x + 10 \leq 16$ .

Possible Answers:

$\left [ -\frac{5}{3}, 2 \right ]$

$\left ( -\frac{5}{3}, 2 \right )$

$\left ( -\frac{5}{3}, 2 \right ]$

$\left [ \frac{5}{3}, 2\right )$

$\left ( \frac{5}{3}, 2 \right ]$

Correct answer:

$\left ( -\frac{5}{3}, 2 \right ]$

Explanation:

$5 < 3x + 10 \leq 16$

Subtract 10: $-5 < 3x \leq 6$

Divide by 3: $-5/3 < x \leq 2$

We must carefully check the endpoints. is greater than $-\frac{5}{3}$ and cannot equal $-\frac{5}{3}$ , yet CAN equal 2. Therefore $-\frac{5}{3}$ should have a parentheses around it, and 2 should have a bracket: is in

$\left ( -\frac{5}{3}, 2 \right ]$

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Example Question #3 : Solving Inequalities

Solve -2x+3<7 .

Possible Answers:

$(-2, \infty )$

$(- \infty , -2)$

$(2, \infty )$

$[-2, \infty ]$

$[-2, \infty )$

Correct answer:

$(-2, \infty )$

Explanation:

-2x+3<7

Subtract 3 from both sides: -2x<4

Divide both sides by : x>-2

Remember: when dividing by a negative number, reverse the inequality sign!

Now we need to decide if our numbers should have parentheses or brackets. is strictly greater than , so should have a parentheses around it. Since there is no upper limit here, is in $(-2, \infty )$ .

Note: Infinity should ALWAYS have a parentheses around it, NEVER a bracket.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1181 : Gmat Quantitative Reasoning

Example Question #1181 : Problem Solving Questions

Example Question #1183 : Gmat Quantitative Reasoning

Example Question #81 : Understanding Exponents

Example Question #81 : Exponents

Example Question #82 : Exponents

Example Question #84 : Understanding Exponents

Example Question #1 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #3 : Solving Inequalities

All GMAT Math Resources

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