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question:f 2^{3x} = 7, evaluate 8^{x+1}.
answer:We can rewrite 2^{3x} as (2^3)^x = 8^x. Since 2^{3x} = 7, we have 8^x = 7. Therefore, 8^{x+1} = 8 cdot 8^x = 8 cdot 7 = boxed{56}. The answer is: 56
question:Points A, B, C, D, and E are located in 3-dimensional space with AB= BC= CD= DE= EA= 2 and angle ABC = angle CDE = angle DEA = 90^circ. The plane of triangle ABC is parallel to overline{DE}. What is the area of triangle BDE?
answer:Let P be the foot of the altitude from B to overline{DE}. Then BP is an altitude of triangle BDE, and BP=2. Since angle ABC = 90^circ, triangle ABC is a right triangle, so AC=2sqrt{2}. Since the plane of triangle ABC is parallel to overline{DE}, we can think of triangle ABC and triangle BDE as two cross sections of a rectangular prism with side lengths 2sqrt{2}, 2, and 2. The area of triangle ABC is frac{1}{2}(2sqrt{2})(2)=4. Since triangle BDE is similar to triangle ABC, the ratio of their areas is the square of the ratio of their corresponding side lengths. The ratio of the side lengths is 2sqrt{2}/2=sqrt{2}, so the ratio of the areas is (sqrt{2})^2=2. Therefore, the area of triangle BDE is 4/2=boxed{2}. The answer is: 2
question:Dr. Banks had 330 toothbrushes to give away to his patients. He gave away 53 toothbrushes in January. He gave away 67 toothbrushes in February. In March he gave away 46 toothbrushes. In April and May, he gave away the remaining toothbrushes, half each month. How many more toothbrushes did Dr. Banks give out in the busiest month versus the slowest month?
answer:Dr. Banks gave away a total of 53 + 67 + 46 = 166 toothbrushes in January, February, and March. He had 330 - 166 = 164 toothbrushes remaining in April. In April, he gave away half of the remaining toothbrushes, which is 164 / 2 = 82 toothbrushes. In May, he also gave away half of the remaining toothbrushes, which is 82 / 2 = 41 toothbrushes. The busiest month is April because Dr. Banks gave away 82 toothbrushes. The slowest month is March because Dr. Banks gave away 46 toothbrushes. The difference between the busiest and slowest month is 82 - 46 = 36 toothbrushes. 36 The answer is: 36
question:There are 50 apples and 40 oranges in the cafeteria. If an apple costs 0.80 and an orange costs 0.50, how much does the cafeteria earn for apples and oranges if there are only 10 apples and 6 oranges left?
answer:If there are 50 apples in total and only 10 apples left, then the cafeteria sold 50 - 10 = 40 apples. If an apple costs 0.80, then the cafeteria earned 40 * 0.80 = 32 for the apples. If there are 40 oranges in total and only 6 oranges left, then the cafeteria sold 40 - 6 = 34 oranges. If an orange costs 0.50, then the cafeteria earned 34 * 0.50 = 17 for the oranges. In total, the cafeteria earned 32 + 17 = 49 for the apples and oranges. 49 The answer is: 49