Exponents and Roots Quiz posted on March 8, 2016 This is a set of GMAT PS and DS problems concerning exponents and roots. If you struggle with these problems then you might want to take a content quiz on the fundamentals of exponents and roots. You have 30 mins to complete the 15 questions Exponents and Roots Please wait while the activity loads. If this activity does not load, try refreshing your browser. Also, this page requires javascript. Please visit using a browser with javascript enabled. If loading fails, click here to try again Start Congratulations - you have completed Exponents and Roots. You scored %%SCORE%% out of %%TOTAL%%. Your performance has been rated as %%RATING%% Your answers are highlighted below. Question 1If \(a = 0.567\), \(b = {(0.567)}^2,\)and \(c =\sqrt{0.567}\), which of the following is the correct ordering of \(a\), \(b,\) and \(c\)Aa < b < cBb < a < cCc < a < bDa < c < bEb < c < aQuestion 1 Explanation: If 0 < A < 1, roots and powers of A are counter intuitive. For numbers in this range roots are larger and exponents are smaller. Consequently, a < c, and b < a. Therefore, b < a < c. The correct answer is B. Question 2\(\sqrt{30 + 42} = \)A\(6\sqrt{5 + 7}\)B\(12\sqrt{3}\)C\(8\)D\(6\sqrt{2}\)E\(3\sqrt{2}\)Question 2 Explanation: I hope this was a pretty straightforward computation: \(\sqrt{30 + 42} = \sqrt{72} = \sqrt{(36)(2)} = \sqrt{36}\times{\sqrt{2}} = 6\sqrt{2}\). The correct answer is D.Question 3Is \(a>b\)? (1) \(a + b\) is postitive. (2) \({b}^a\) is negative.AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.BStatement (2) ALONE is sufficient , but statement (1) alone is not sufficient to answer the question asked.CBOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.DEACH statement ALONE is sufficient to answer the question asked.EStatements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.Question 3 Explanation: The fact that the sum of \(a\) and \(b\) is positive doesn't tell us anything about the relative values of \(a\) and \(b\). 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 4If \(n=\sqrt{\frac{81}{16}}\), then what is the value of \(\sqrt{n}\)?A\(\frac{2}{9}\)B\(\frac{3}{2}\)C\(\frac{9}{4}\)D\(4\)E\(9\)Question 4 Explanation: Recall the rule \(\sqrt{\frac{A}{B}} = \frac{\sqrt{A}}{\sqrt{B}}\). 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 5If \(x\) is a positive integer and \(A = 3.2\times{10}^x\), what is the value of \(x\)?(1) \(40,000< A < 3,000,000\).(2) \(\sqrt[4]{A}=20\sqrt[4]{2}.\)AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.BStatement (2) ALONE is sufficient , but statement (1) alone is not sufficient to answer the question asked.CBOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.DEACH statement ALONE is sufficient to answer the question asked.EStatements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.Question 5 Explanation: First, convert all values to scientific notation: \(40,000 = {4.0}\times{{10}^4}\) and \(3,000,000 = {3.0}\times{{10}^6}\). If \(x = 4\), then \(A\) is less than \(40,000\), and if \(x = 6\), then \(A\) is greater than \(3,000,000\). 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 6If \(n\) is an integer, and \({0.5}^{n} < 0.001\), which of the following could be the value of \(n\)?A\(-10\)B\(-2\)C\(0\)D\(2\)E\(10\)Question 6 Explanation: I tell all of my students that it pays to know powers of two. When bacteria grow, or grasshoppers multiply, or radio isotopes decay on the GMAT, they almost always double or get cut in half. 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 \(\frac{1}{1024} < \frac{1}{1000}[/latex]. The correct answer is E.Question 7Is [latex]x\) an integer? (1) \({x}^3\) is an integer. (2) \(\sqrt[3]{x}\) is an integer.AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.BStatement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.CBOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.DEACH statement ALONE is sufficient to answer the question asked.EStatements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.Question 7 Explanation: Statement (1) is insufficient. Consider the following examples \(x = \sqrt[3]{2}\), and \(x = 2\). In the former \(x\) is not an integer, but \({x}^3\) is an integer. In the latter example \(2\) is an integer and \({2}^3\) is also an integer. 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 8Does \( x = y \)?(1) \({A}^x = {A}^y\).(2) \(A\) is a positive integer.AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.BStatement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.CBOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.DEACH statement ALONE is sufficient to answer the question asked.EStatements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.Question 8 Explanation: Statement (1) is insufficient. 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 9\(\frac{(0.2)^5}{(0.2)^2}=\)A\(0.001\)B\(0.002\)C\(0.008\)D\(0.02\)E\(0.3\)Question 9 Explanation: When we divide exponential expressions with the same base we take the difference of the exponents and keep the base: \(\frac{A^x}{A^y} = A^{x-y}\). If we apply this rule to the problem, we get \((0.2)^{5-2} = (0.2)^3 = (0.2)(0.2)(0.2) = 0.008\). The correct answer is C.Question 10What is the value of \(x\)?(1) The seventh root of \(x\) is \(3.7\).(2) \(9493 < x < 9495[/latex].AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.BStatement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.CBOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.DEACH statement ALONE is sufficient to answer the question asked.EStatements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.Question 10 Explanation: Statement (1) is sufficient. Given some time and/or a calculator we COULD find [latex]{3.7}^7\). 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 11If \(n\) is an integer, is \({(0.5)}^n>{2}^n\)? (1) \(n < 5\) (2) \(n > -5\)AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.BStatement (2) ALONE is sufficient , but statement (1) alone is not sufficient to answer the question asked.CBOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.DEACH statement ALONE is sufficient to answer the question asked.EStatements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.Question 11 Explanation: Statements (1) and (2) alone allow for an infinite number of possible values for \(n\). 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. Question 12\({(\sqrt{5}+3\sqrt{5})}^2=\)A\(20\)B\(16\sqrt{5}\)C\(50\)D\(80\)E\(100\)Question 12 Explanation: Recall the rule \(A\sqrt{x}+B\sqrt{x}=(A+B)\sqrt{x}\). So, \(\sqrt{5}+3\sqrt(5)=4\sqrt{5}\). Therefore, \({(4\sqrt{5})}^2=16(5)=80\). The correct answer is D.Question 13\(\sqrt{{(2)(18)}+{(12)(9)}}\)A\(\sqrt{20}\)B\(6\)C\(8\)D\(\sqrt{20}+2\sqrt{2}\)E12Question 13 Explanation: You can take a brute force approach: \(\sqrt{36+108} = \sqrt{144} = 12.\) 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 14\(\sqrt{63}+\sqrt{28}=\)A\(5\sqrt{7}\)B\(7\sqrt{5}\) C\(10\sqrt{7}\) D\(13\sqrt{7}\)E\(50\)Question 14 Explanation: GOOD ENOUGH, simplify the roots: \(\sqrt{63}+\sqrt{28} = \sqrt{(9)(7)}+\sqrt{(4)(7)} = \sqrt{9}\sqrt{7}+\sqrt{4}\sqrt{7} = 3\sqrt{7}+2\sqrt{7} = 5\sqrt{7}\) 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 AQuestion 15What is the units digit of \({37}^{37} + {55}^{55}\)?A\(0\)B\(2\)C\(4\)D\(6\)E\(8\)Question 15 Explanation: A GMAT Classic! The units digit of any power of \(55\) (or \(555\) or \(1,000,000,005\) or any integer ending in \(5\)) has a units digit of \(5\). The units digit sequence for powers of any integer ending in \(7\) is \(7, 9, 3, 1, 7, 9, 3, 1\), and so on - repeating the same pattern until the heat death of the universe. When we divide \(37\) by \(4\) we get a quotient of \(9\) and a remainder of \(1\): i.e. \(37 = (9)(4) + 1\). So, the patter repeats nine times and travels one more step landing on seven. The sum of an integer ending in \(5\) and an integer ending in \(7\) has a units digit of \(2\). 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