Exponents and Roots
Question 1 
A  a < b < c 
B  b < a < c 
C  c < a < b 
D  a < c < b 
E  b < c < a 
The correct answer is B.
Question 2 
(1) The seventh root of \(x\) is \(3.7\).
(2) \(9493 < x < 9495[/latex].
A  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. 
B  Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked. 
C  BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. 
D  EACH statement ALONE is sufficient to answer the question asked. 
E  Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed. 
Given some time and/or a calculator we COULD find [latex]{3.7}^7\).
Statement (2) is not sufficient. It would be sufficient if we knew that \(x\) is an integer, but no such luck.
The correct answer is A.
Question 3 
(1) \({A}^x = {A}^y\).
(2) \(A\) is a positive integer.
A  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. 
B  Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked. 
C  BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. 
D  EACH statement ALONE is sufficient to answer the question asked. 
E  Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed. 
Clearly, if \({2}^x = {2}^y\) ,then \(x = y\), but there are several scenarios where \({A}^x = {A}^y\) does not mean that \(x = y\). For instance if \(A = 1\), then \(x\) and \(y\) can take any value because 1 to any power is always 1.
Statement (2) doesn't eliminate the two possibilities mentioned above as 1 and 2 are both positive integers. Further, statement (2) doesn't give us any information that we can combine with statement (1), so statements (1) and (2) together are insufficient.
The correct answer is E.
Question 4 
A  \(\frac{2}{9}\) 
B  \(\frac{3}{2}\) 
C  \(\frac{9}{4}\) 
D  \(4\) 
E  \(9\) 
So, we get, \(n = \sqrt{\frac{81}{16}} = \frac{\sqrt{81}}{\sqrt{16}} = \frac{9}{4}\).
And don't forget that you need to take the square root of \(n\): \(\frac{\sqrt{9}}{\sqrt{4}}=\frac{3}{2}\).
The correct answer is B.
Question 5 
A  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. 
B  Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked. 
C  BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. 
D  EACH statement ALONE is sufficient to answer the question asked. 
E  Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed. 
A little algebra makes it clear that statement (2) is sufficient :
\(\sqrt[3]{x} = integer\)
\({\sqrt[3]{x}}^3 = {integer}^3\)
\(x = {integer}^3\).
So, \(x\) is the cube of an integer and is clearly an integer.
The correct answer is B.
Question 6 
A  \(6\sqrt{5 + 7}\) 
B  \(12\sqrt{3}\) 
C  \(8\) 
D  \(6\sqrt{2}\) 
E  \(3\sqrt{2}\) 
\(\sqrt{30 + 42} = \sqrt{72} = \sqrt{(36)(2)} = \sqrt{36}\times{\sqrt{2}} = 6\sqrt{2}\). The correct answer is D.
Question 7 
A  \(0\) 
B  \(2\) 
C  \(4\) 
D  \(6\) 
E  \(8\) 
Question 8 
A  \(0.001\) 
B  \(0.002\) 
C  \(0.008\) 
D  \(0.02\) 
E  \(0.3\) 
\(\frac{A^x}{A^y} = A^{xy}\).
If we apply this rule to the problem, we get \((0.2)^{52} = (0.2)^3 = (0.2)(0.2)(0.2) = 0.008\).
The correct answer is C.
Question 9 
A  \(\sqrt{20}\) 
B  \(6\) 
C  \(8\) 
D  \(\sqrt{20}+2\sqrt{2}\) 
E  12 
Or you can get fancy:\(\sqrt{9(4+12)} = \sqrt{(9)(16)} = \sqrt{9}\sqrt{16} = (3)(4) = 12\).
The correct answer is E.
Question 10 
(1) \(40,000< A < 3,000,000\).
(2) \(\sqrt[4]{A}=20\sqrt[4]{2}.\)
A  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. 
B  Statement (2) ALONE is sufficient , but statement (1) alone is not sufficient to answer the question asked. 
C  BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. 
D  EACH statement ALONE is sufficient to answer the question asked. 
E  Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed. 
So, we know \(x = 5\). Thus, statement (1) is sufficient. Statement (2) may look a bit intimidating, however, this is a data SUFFICIENCY question, and given some time and/or a calculator, we COULD find \(A\). So, statement (2) is also sufficient.
The correct answer is D.
Question 11 
(1) \(a + b\) is postitive.
(2) \({b}^a\) is negative.
A  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. 
B  Statement (2) ALONE is sufficient , but statement (1) alone is not sufficient to answer the question asked. 
C  BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. 
D  EACH statement ALONE is sufficient to answer the question asked. 
E  Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed. 
So statement (1) is insufficient. If an exponential expression is negative we know that the base must be negative and the exponent odd. However, we still don't know anything about the relative values of \(a\) and \(b\). For example, both \({2}^{3}\) and \({2}^3\) are negative: \(2\) and \(8\), respectively. In the former case 3 < 2, so a < b. In the latter case 3 > 2, so a > b.
Combining the two statements, we know that \(b\) must be negative, and because the sum of \(a\) and \(b\) must t be positive, \(a\) is positive. Therefore, \(a>b\).
The correct answer is C.
Question 12 
A  \(10\) 
B  \(2\) 
C  \(0\) 
D  \(2\) 
E  \(10\) 
I recommend knowing powers of two up to \({2}^{10}\). Here it pays of in a different context.
\(0.001 = \frac{1}{1000}\) and \({2}^{10} = 1024\). And \(\frac{1}{1024} < \frac{1}{1000}[/latex].
The correct answer is E.
Question 13 
A  \(20\) 
B  \(16\sqrt{5}\) 
C  \(50\) 
D  \(80\) 
E  \(100\) 
So, \(\sqrt{5}+3\sqrt(5)=4\sqrt{5}\).
Therefore, \({(4\sqrt{5})}^2=16(5)=80\).
The correct answer is D.
Question 14 
A  \(5\sqrt{7}\) 
B  \(7\sqrt{5}\)

C  \(10\sqrt{7}\)

D  \(13\sqrt{7}\) 
E  \(50\) 
BETTER, use estimation: \(63\approx{64}\) and \(28\approx{25}\), so \(\sqrt{63}+\sqrt{28} \approx\sqrt{64}+\sqrt{25} = 8+5 = 13\). Estimation is often the most efficient way to solve a problem because numerical answers to GMAT questions are always in order from least to greatest or greatest to least. Here it's clear the answer choices B, C, D, and E are all larger than 13, so the correct answer is A
Question 15 
(1) \(n < 5\)
(2) \(n > 5\)
A  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. 
B  Statement (2) ALONE is sufficient , but statement (1) alone is not sufficient to answer the question asked. 
C  BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. 
D  EACH statement ALONE is sufficient to answer the question asked. 
E  Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed. 
In particular \(n = 1\) and \(n = 1\) work for both statements. If \(n = 1\) the answer to the question is NO. If \(n = 1\) the answer to the question is YES.
The correct answer is E.
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