All GMAT Math Resources
Example Questions
Example Question #1 :Dsq: Understanding Powers And Roots
is a real number. Ispositive, negative, or zero?
Statement 1:
Statement 2:
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.
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.
EITHER statement ALONE is sufficient to answer the question.
If, then, and, somustbe positive.
If, then,. and, so again,mustbe positive. Either statement is enough to answer the question in the affirmative.
Example Question #1 :Powers & Roots Of Numbers
Simplify this expression as much as possible:
The expression is already simplified
Example Question #3 :Powers & Roots Of Numbers
Imagine an integersuch that the units digit ofis greater than 5. What is the units digit of?
(1) The units digit ofis the same as the units digit of.
(2) The units digit ofis the same as the units digit of.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
BOTH statements TOGETHER are not sufficient to answer the question.
EACH statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
(1) The only single-digit integer greater than 5 whose unit digit of its square term is equal to itself is 6. This statement is sufficient.
(2) There are two single-digit integers where the unit digit of the cubed term is equal to the integer itself: 6 and 9. This statement is insufficient.
Example Question #1 :Powers & Roots Of Numbers
What is the value of twelve raised to the fourth power?
"Twelve raised to the fourth power" is 124. If you can translate the words into their mathematical counterpart, you're done, because the actual calculation should be done by your calculator. It will tell you that. There is not enough time on the test for you to try to do this by hand.
Example Question #5 :Powers & Roots Of Numbers
Calculate the fifth root of:
(1) The square root ofis.
(2) The tenth root ofis.
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 sufficient, but NEITHER statement ALONE is sufficient.
Both statements TOGETHER are not sufficient.
Each statement ALONE is sufficient.
Each statement ALONE is sufficient.
Using Statement (1):
Statement (1) ALONE is SUFFICIENT.
Using Statement (2):
Statement (2) ALONE is SUFFICIENT.
Therefore EACH Statement ALONE is sufficient.
Example Question #6 :Powers & Roots Of Numbers
is a positive real number. True or false:是一个理性的number.
Statement 1:is an irrational number.
Statement 2:is an irrational number.
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.
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.
EITHER statement ALONE is sufficient to answer the question.
一个integer power of a rational number, being a product of rational numbers, must itself be rational. Either statement alone asserts that such a power is irrational, so conversely, either statement alone provesirrational.
Example Question #7 :Powers & Roots Of Numbers
. True or false:是理性的。
Statement 1:是理性的。
Statement 2:是理性的。
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.
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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 alone is not enough to proveis or is not rational. Examples:
If, then
If, then
In both cases,is rational, but in one case,is rational and in the other,is irrational.
A similar argument demonstrates Statement 2 to be insufficient.
Assume both statements are true.andare rational, so their difference is as well:
is rational, so by closure under division,是理性的。
Example Question #1 :Powers & Roots Of Numbers
. True or false:是理性的。
Statement 1:is irrational.
Statement 2:是理性的。
EITHER 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.
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.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 alone is not enough to proverational or irrational. Examples:
If, then
If, then
In both cases,is irrational, but in only one case,是理性的。
Assume Statement 2 alone.is rational, so, by closure of the rational numbers under multiplication,
是理性的。The rationals are closed under addition, so the sum
是理性的。
Example Question #9 :Powers & Roots Of Numbers
is a positive real number. True or false:是一个理性的number.
Statement 1:is irrational.
Statement 2:is irrational.
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.
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.
Ifis rational, then, since the product of two rational numbers is rational,是理性的。If Statement 1 alone is assumed, then, sinceis irrational,must be irrational.
Assume Statement 2 alone, and note that
In other words,is the square root of. Since both rational and irrational numbers have irrational square roots,being irrational does not prove or disprove that是理性的。
Example Question #10 :Powers & Roots Of Numbers
is a positive real number. True or false:是一个理性的number.
Statement 1:是一个理性的number.
Statement 2:是一个理性的number.
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.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 alone provides insufficient information.is a number with a rational cube root,, and a rational square root,.is a number with a rational cube root,, but an irrational square root.
Now assume Statement 2 alone.
In other words,is the square of. The rational numbers are closed under multiplication, so ifis rational,是理性的。