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Solution
234
Rewrite the numerator in terms of powers of 3. 81 = 34, so 813 = (34)2, or 312. 27 = 33, so 273 = (33)3, or 39.Therefore, you can rewrite the entire numerator as \(\frac{3^{12}-3^{9}}{3^{7}}\). Now you can factor the numerator so that you get \(\frac{3^{9}(3^{3}-1)}{3^{7}}\)=32(26)=234.
An empty, cube-shaped swimming pool is filled part way with x cubic feet of water. It is then filled the rest of the way with y cubic feet of chlorine. Which of the following, in feet, expresses the depth of the swimming pool?
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Solution
Use Plugging In to solve the problem.The swimming pool has a total volume of (x + y).You’re trying to find the depth, or one side of the cube.Choose easy numbers. It helps to start with the depth, which is your target. If the depth is 2, then the total volume has to be 23, or 8.You could choose x = 7 and y = 1, but really you only use x + y in the answers, so all you need is x + y = 8. Now Plug In to find your target in the choices.Choice (A) = 8, which doesn’t match.Choice (B) is a fraction, which doesn’t match.Choice (C) is \(\sqrt[3]{8}\) which does equal 2, so keep it.Choice (D) is 83, which doesn’t match.Choice (E) is 2⁄3 which doesn’t match.
\(\left ( \sqrt{79}-1 \right )\left ( \sqrt{79}+1 \right )\left ( \sqrt{78}-1 \right )\left ( \sqrt{78}+1 \right )=\)
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Solution
6006
You can try to hammer this out on your calculator, but it’s a lot easier to use the common quadratic (x − y)(x + y) = x2 − y2. Start with the first 2 terms:\(\left ( \sqrt{79}-1 \right )\left ( \sqrt{79}+1 \right )=\left ( \sqrt{79} \right )^{2}-1^{2}\), or 79 − 1 = 78. For the last 2 terms,\(\left ( \sqrt{79}-1 \right )\left ( \sqrt{79}+1 \right )=\left ( \sqrt{79} \right )^{2}-1^{2}\), or 78 − 1 = 77.The whole expression, then, equals 78 × 77 = 6006. If you don’t recognize the common quadratic, you can get the same product by FOILing the first 2 terms and the last 2 terms separately and multiplying the results.
If j is a nonzero integer, which of the following must be greater than j?
Indicate \(\underline{all}\) possible values.
| A.j–2 |
| B.j–1 |
| C.j0 |
| D.j2 |
| E. |
| F.j4 |
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Solution
and F As soon as you see variables in the answer choices, set up your scratch paper to Plug In. Start with an easy number like j = 2; choices (A) and (B) are fractions and choice (C) is 1, so eliminate all three. Next, try a number like j = –2; now choice (E) is –8, so eliminate it.Try more numbers if time permits; choices (D) and (F) will always work.
If pq ≠ 0, and 1⁄p = √q , what is the value of p ?
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Solution
As soon as you see variables in the answer choices, set up your scratch paper to Plug In. Start with the number under the radical: If q = 4, then 1⁄q = 2, and p, which is also your target answer, is 1⁄2. Plug 4 into the answers for q, and only choice (C) is 1⁄2.
If \(\sqrt[3]{x+3}=4,x=\)
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Solution
61
Raise both sides of the equation to the third power, and you’ll have x + 3 = 64, so x = 61.
Which of the following expressions is equivalent to 17,640?
Indicate \(\underline{all}\) such expressions.
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Solution
A,C, and D
Based on the answer choices, it looks like you’re being asked to find the prime factors of 17,640 and then re-write them in a few different ways. Instead of starting there, though, take a look at the number you’re being asked to factor.Clearly, it’s a multiple of 10. And if it’s a multiple of 10, then, whatever else might factor in, a 5 and a 2 have to show up somewhere.Eliminate choices (B) and (E), neither of which contains a 5. From there, look for an opportunity to use your on-screen calculator easily:Choice (D) shouldn’t be too hard to multiply (as there are no exponents) and works out to 17,640.Expand out the 8 (2 × 2 × 2) and the 9 (3 × 3) of choice (D) to compare to choice (A).They are equivalent. Finally, you may either use the on-screen calculator to check choice (C), or simply compare to choice (A) (they’ve combined a 2 and a 5, and compressed the remaining numbers since they all have the same power). In either case, you should get that it also works out to 17,640.
Which of the following is equal to \((\sqrt[3]{64}+\sqrt[3]{8})^{2}\)?
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Solution
First solve for the cube roots of 64 and 8.\(\sqrt[3]{64}\) = 4, and \(\sqrt[3]{64}\) = 2. Next, calculate the value inside the parentheses.You can now rewrite the equation as (4 + 2)2 = (6)2 = 36.The correct answer is choice (C).
y ≠ 0
| Quantity A | Quantity B |
| \(-\frac{y^{3}}{2}\) | \(\frac{y^{2}}{2}\) |
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Solution
D
Plug in for y. If y = 2, then in Quantity A you have \(-\frac{2^{3}}{2}\) = –4, and in Quantity B you have \(-\frac{2^{3}}{2}\) = 2. In this case, Quantity B is bigger than Quantity A, so you can eliminate choices (A) and (C). Plug in again using y = –2 : in Quantity A you have \(\frac{(-2)^{3}}{2}\) = 4, and in Quantity B you have \(\frac{(-2)^{3}}{2}\) = 2. In this case, Quantity A is bigger, so you can eliminate choice (B).The correct answer is therefore choice (D).
If \(\sqrt[3]{\frac{x^{\frac{3}{4}}}{x^{-\frac{13}{4}}}}=(x^{\frac{1}{4}}\cdot x^{\frac{5}{4}})^{2}\) then x =
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Solution
3
Here’s how to simplify this equation using laws of exponents:
\(\sqrt[3]{\frac{x^{\frac{3}{4}}}{x^{-\frac{13}{4}}}}=(x^{\frac{1}{4}}\cdot x^{\frac{5}{4}})^{2}\)
\(\sqrt[3]{\frac{x^{\frac{3}{4}}}{x^{-\frac{13}{4}}}}=(x^{\frac{1}{4}}\cdot x^{\frac{5}{4}})^{2}\)
\(3\sqrt{x^{\frac{16}{4}}}=x^{3}\)
\(3\sqrt{x^{4}}=x^{3}\)
3x2 = x2
3 = \(\frac{x^{3}}{x^{2}}\)
3 = x