If −1 < a < 0, q = a − 1, r = a2, and s = a3, then which of the following is true?
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Solution
Plug in values for y. If y = 1, then Quantity A is 8 and Quantity B is 9. In this case, Quantity B is larger, so eliminate choices (A) and (C). If 1⁄2, then Quantity A is 64 and Quantity B is 36; eliminate choice (B).You are left with choice (D).
If \(b=\frac{c+d^{2}}{c}\) and \(a=\frac{c^{2}}{d^{2}}\), what is b in terms of a ?
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Solution
\(\sqrt[3]{69}\) means “the number that when you cube it, gives you 69”. So plug in the answer choices, cubing each one until you find the value closest to 69. It is easier to start with the smaller values first. 33 = 27; 43 = 64; 53 = 125.Clearly, 43 is closest to 69, so the answer is choice (D)
| Quantity A | Quantity B |
| 22(420) | 25(105) |
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Solution
In Quantity A, evaluate the root first, then attach the minus sign:-√9 = -3; this is equivalent to Quantity B:\(\sqrt[3]{-27}\) Thus, the answer is choice (C).
| Quantity A | Quantity B |
| (y − x)7(y − x)2 |
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Solution
Evaluate the relationship between the quantities by plugging in values for x. If x = 2, then Quantity A is 1⁄9 and Quantity B is 1⁄4; Quantity B is greater, so eliminate answer choices (A) and (C). Now, if x = 3, then Quantity A is \(\frac{1}{27}\) and Quantity B is − 1⁄8; Quantity A is now greater, so eliminate choice (B), and you’re left with choice (D).
What is the value of x2 – 1 when 9x + 1 = 27x + 1 ?
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Solution
First, add the numbers under the root symbol.To simplify \(\sqrt{90}\) factor out the perfect square 9.\(\sqrt{90}\) = √9 + \(\sqrt{10}\) = 3\(\sqrt{10}\).The answer is choice (C).
If √x = 4, then x2 =
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Solution
Quantity A, (0.5)3, equals 0.125. Quantity B,(5)3(1⁄2)3(1⁄5)3, equals(125)(1⁄8)(\(\frac{1}{125}\)); the first and third terms cancel to leave only 1⁄8=0.125.The quantities are equal. Alternatively, if you’re comfortable with your exponent rules, you can combine and cancel the terms in Quantity B, and compare the quantities without calculating either one:(0.5)3, equals 0.125. Quantity B,(5)3(1⁄2)3(1⁄5)3 = (5 × 1⁄2 × 1⁄5)3. Again, the first and third terms cancel to leave only (1⁄2)3, which is the same as (0.5)3.
(√5 + √7)2 =
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Solution
First, take the square root of 64, which is 8.√8 = √4 × √2 = 2√2 .Choice (A) is correct.
| Quantity A | Quantity B |
| (x3 + 1)2 | x6 |
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Solution
Simplify each of the expressions by subtracting the exponents.You get 510 in Quantity A and 512 in Quantity B.
| Quantity A | Quantity B |
| \(\sqrt{\frac{1}{4^{2}}}\) | 1⁄4 |
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Solution
Plug In! If x = 2 and y = 3, then Quantity A is 5 and Quantity B is 25. Quantity B is greater, so eliminate choices (A) and (C). Next, make x and y both 0.Both Quantities A and B are now 0, thus, they are equal.Eliminate choice (B), and you’re left with choice (D).
\(\left( \frac{9^{2}-3^{2}}{6^{2}} \right)\) =
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Solution
Although it may be tempting to try some fancy factoring, this problem is more easily solved by calculating the individual exponential expressions:\(\frac{9^{2}-3^{2}}{6^{2}}=\frac{81-9}{36}=\frac{72}{36}=2\).Be careful if you got choice (B):You may have incorrectly subtracted 92 – 32 in the numerator and gotten 62.