GMAT Math : DSQ: Understanding exponents

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Example Question #1 : Algebra

Which is the greater quantity, $3^{x}$ or $9^{y}$ - or are they equal?

Statement 1: x = 2y + 2

Statement 2: x = 3y

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

From Statement 1 alone,

$3^{x} = 3^{2y+2} = 3^{2y} \cdot 3^{2} =\left ( 3^{2} \right ) ^{y}\cdot 9 = 9^{y} \cdot 9>9^{y}$

Now assume Statement 2 alone. We show that this is insufficient with two cases:

Case 1: x = 3,y= 1

$3^{3} = 27$ ; $9^{1} = 9$ ; therefore, $3^{x} >9^{y}$

Case 1: x = -3,y=- 1

$3^{-3} = \frac{1}{27}$ ; $9^{-1} = \frac{1}{9}$ ; therefore, $3^{x}<9^{y}$

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Example Question #2 : Algebra

Does $\log_{2} \left ( MN \right )$ exist?

Statement 1: and are both negative.

Statement 2: divided by 2 yields an integer.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

A logarithm can be taken of a number if and only if the number is positive. If Statement 1 alone is true, then , being the product of two negative numbers, must be positive, and $\log_{2} \left ( MN \right )$ exists.

Statement 2 is irrelevant; 4 and both yield integers when divided by 2, but $\log_{2} 4 = 2$ and $\log_{2}\left ( -4 \right )$ does not exist.

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Example Question #3 : Algebra

Johnny was assigned to write a number in scientific notation by filling the circle and the square in the pattern below with two numbers.

$\bigcirc \times 10 ^{\square }$

Johnny filled in both shapes with numbers. Did he succeed?

Statement 1: He filled in the circle with the number "10".

Statement 2: He filled in the square with a negative integer.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The number $a \times 10 ^{n}$ is a number written in scientific notation if and only of two conditions are true:

1) $1 \leq \left | a \right | < 10$

2) is an integer

By Statement 1 Johnny filled in the circle incorrectly, since it makes a = 10 .

By Statement 2, Johnny filled in the square correctly, but the statement says nothing about how he filled in the circle; Statement 2 leaves the question open.

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Example Question #4 : Algebra

Is $x^{2} < x$ ?

(1) 0 < x < 1

(2) x > 0

Possible Answers:

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

E: Statements (1) and (2) TOGETHER are not sufficient

D: EACH statement ALONE is sufficient

Correct answer:

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

Explanation:

$x^{2} < x \Leftrightarrow x^{2}-x<0 \Leftrightarrow x(x-1)<0$

From statement 1 we get that x > 0 and -1 < x - 1 < 0 .

So the first term is positive and the second term is negative, which means that x (x-1) is negative; therefore the statement 1 alone allows us to answer the question.

Statement 2 tells us that x > 0 . If x = 0.5 , we have $x^{2}=0.25$ which is less than 0.5 . Therefore in this case $x^{2}<x$ .

For x = 2 , we have $x^{2} = 4$ which is greater than . So in this case $x^{2} > x$ .

So statement 2 is insufficient.

Therefore the correct answer is A.

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Example Question #5 : Algebra

Solve the following rational expression:

$\frac{x^{m-n}10^{n-1}}{10^{(\frac{m}{2}-2)}x^{n}}$

(1) m=5

(2) m=2n

Possible Answers:

EACH statement ALONE is sufficient to answer the question

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient

Both statements TOGETHER are not sufficient.

Correct answer:

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

Explanation:

When replacing m=5 in the expression we get:

$\frac{x^{5-n}10^{n-1}}{10^{\frac{1}{2}}x^{n}}$

Therefore statement (1) ALONE is not sufficient.

When replacing m=2n in the expression we get:

$\frac{x^{2n-n}10^{n-1}}{10^{(\frac{2n}{2}-2)}x^{n}}=\frac{x^{n}10^{n-1}}{10^{n-2}x^{n}}=\frac{10^{n-1}}{10^{n-2}}=10$

Therefore statement (2) ALONE is sufficient.

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Example Question #6 : Algebra

Myoshi has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.

$\bigcirc \times 10^{\square }$ .

Did Myoshi succeed?

Statement 1: Myoshi wrote in the circle.

Statement 2: Myoshi wrote in the square.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

A number in scientific notation takes the form

$a \times 10 ^{n}$

where $1 \le |a| < 10$ and is an integer of any sign.

Assuming Statement 1 alone, Myoshi did not succeed, since she entered an incorrect number into the circle - $|-24| = 24 \ge 10$ .

Statement 2 alone is inconclusive. Myoshi entered a correct number into the square, since is an integer. But the question is open, since it is not known whether she entered a correct number into the circle or not.

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Example Question #7 : Algebra

is a nonzero number. Is it negative or positive?

Statement 1: $N \ge N^{2}$

Statement 2: $N >N^{3}$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

All negative numbers are less than their (positive) squares, as are all positive numbers greater than 1. Therefore, if Statement 1 is assumed, $N \in (0,1]$ .

can be determined to be positive.

Statement 2 alone is inconclusive. For example, if $N = \frac{1}{2}$ , then $N^{3} = \frac{1}{8}$ , and if N = -2 , $N^{3} = -8$ . In both cases, $N >N^{3}$ , but has different signs in the two cases.

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Example Question #8 : Algebra

is a nonzero number. Is it negative or positive?

Statement 1: $N < N^{2}$

Statement 2: $N < N^{3}$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements together are inufficient to produce an answer. For example,

If N = 2 , then $N^{2} = 4$ and $N^{3} = 8$ .

If $N = -\frac{1}{2}$ , then $N^{2} = \frac{1}{4}$ and $N^{3} = -\frac{1}{8}$ .

In both cases, $N < N^{2}$ and $N < N^{3}$ , but the signs of differ between cases.

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Example Question #9 : Algebra

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x^{4}, x^{5}, x^{6}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right | > 1$

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 alone is inconclusive, as can be demonstrated by examining two negative values of other than .

Case 1: x = -2 .

Then

$x^{4} = (-2)^{4} = 16$

$x^{5} = (-2)^{5} = -32$

$x^{6} = (-2)^{6} = 64$

$x^{6}$ is the greatest of these values.

Case 2: $x = -\frac{1}{2}$

Then

$x ^{4}=\left ( -\frac{1}{2} \right )^{4} = \frac{1}{16}$

$x ^{5}=\left ( -\frac{1}{2} \right )^{5} = -\frac{1}{32}$

$x ^{6}=\left ( -\frac{1}{2} \right )^{6} = \frac{1}{64}$

$x^{4}$ is the greatest of these values.

Now assume Statement 2 alone. Either x > 1 or x < -1 .

Case 1: x > 1 .

Then $x \cdot x^{4} > 1\cdot x^{4}$ , so $x^{5} > x^{4}$ ; similarly, $x^{6} > x^{5}$ .

$x^{6}$ is the greatest of the three.

Case 2: x < -1 .

Odd power $x^{5}$ is negative, and even powers $x^{4}$ and $x^{6}$ are positive, so one of the latter two is the greatest. Since x < -1 , it follows that $x^{2} > 1$ . It then follows that $x^{2} \cdot x^{4} > 1 \cdot x^{4}$ , or $x^{6} > x^{4}$ .

Again, $x^{6}$ is the greatest of the three.

Statement 2 alone is sufficient, but not Statement 1.

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Example Question #801 : Data Sufficiency Questions

Chord

Note: Figure NOT drawn to scale.

Examine the above diagram. True or false: $\overline{AB} \cong \overline{YZ}$ .

Statement 1: $\Delta ACB \sim \Delta ZXY$

Statement 2: $\Delta ACB$ and $\Delta ZXY$ have the same perimeter.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

From Statement 1 alone, it follows by the similarity of the triangles that $\angle C \cong \angle X$ . These are congruent inscribed angles of a circle, which intercept congruent arcs, so $\widehat{AXB} \cong \widehat{YCZ}$ . Since congruent arcs have congruent chords, $\overline{AB} \cong \overline{YZ}$ .

Statement 2 alone only tells us the relative perimeters of the triangles. We have no way of determining the individual sidelengths or angle measures relative to each other, so Statement 2 alone is inconclusive.

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GMAT Math : DSQ: Understanding exponents

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Algebra

Example Question #2 : Algebra

Example Question #3 : Algebra

Example Question #4 : Algebra

Example Question #5 : Algebra

Example Question #6 : Algebra

Example Question #7 : Algebra

Example Question #8 : Algebra

Example Question #9 : Algebra

Example Question #801 : Data Sufficiency Questions

All GMAT Math Resources

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