Math: Exponents, Roots, and Word Problems

Faucet A fills at 1/3 tank per hour, faucet B drains at 1/9 tank per hour. Net fill rate = 1/3 - 1/9 = 2/9 tank per hour. In 1.5 hours: 2/9 × 3/2 = 1/3 tank. Option B is correct.

~4.3 = 4 (nearest integer), ~6.7 = 7, ~(-0.9) = -1. Sum: 4 + 7 + (-1) = 10. Option C is correct.

Hourly speeds: 10, 20, 30, ... mph (arithmetic progression). Total distance = sum of first n terms = n(n+1)/2 × 10 = 5n(n+1). For 450 miles: 5n(n+1) = 450, n(n+1) = 90, n² + n - 90 = 0, (n+10)(n-9) = 0. Positive integer: n = 9 hours. Option C is correct.

Rearranging: a² + 10b² + c² - 6ab - 2bc = 0. This factors as (a-3b)² + (b-c)² = 0. For sum of squares to be zero, each term must be zero: a - 3b = 0 (so a = 3b), and b - c = 0 (so c = b). From these: c = a/3. Statements I, III, and VI are true. Option D is correct.

If xy > 0 (both positive or both negative), then taking reciprocals preserves the inequality: 1/x < 1/y implies x > y. Wait, let me reconsider: if x and y are both positive and 1/x < 1/y, then x > y. If x and y are both negative and 1/x < 1/y, then x > y still holds. But the question asks when x < y. Actually, option C says x < y if xy < 0. If xy < 0, one is positive, one negative. Suppose x is positive, y is negative: then 1/x > 0 > 1/y, so 1/x > 1/y, contradicting 1/x < 1/y. Suppose x is negative, y is positive: then 1/x < 0 < 1/y, so 1/x < 1/y holds, and x < y (negative < positive). So option C is correct.

To find the average football sales for the first 8 months of 2004, we need to calculate the total number of footballs sold in the first 8 months and then divide it by 8.

The average number of footballs sold in the first half of the year (6 months) is 120 per month. So the total number of footballs sold in the first 6 months is 120 * 6 = 720.

In the next 2 months, the average number of footballs sold is 80 per month. So the total number of footballs sold in the next 2 months is 80 * 2 = 160.

To find the average for the first 8 months, we add the total number of footballs sold in the first 6 months and the next 2 months, and then divide by 8.

Total number of footballs sold = 720 + 160 = 880.

Average football sales for the first 8 months = 880 / 8 = 110.

Therefore, the correct answer is E. 110.

When percentages are applied successively, they compound. After 15% increase: new price = 1.15 × original. After 40% increase on this: final price = 1.40 × 1.15 × original = 1.61 × original. This is a 61% total increase. Common mistakes include adding the percentages (15% + 40% = 55%) or using simple multiplication (15% × 40% = 6%).

First multiply: 0.003 × 0.07 = 0.00021. To convert to percentage, multiply by 100: 0.00021 × 100 = 0.021%. Common errors come from miscounting decimal places - option A forgets to multiply by 100, option C shifts one place, option D shifts two places, and option E shifts three places.

11^3 = 11 × 11 × 11 = 1331. This can be calculated as 11 × 11 = 121, then 121 × 11 = 1331. The pattern of cubes ending in 1 is useful: 1^3=1, 11^3=1331, 21^3=9261.

The last digit of powers of 2 follows a repeating cycle of 4: 2, 4, 8, 6. For 2^18, we find the position in the cycle: 18 ÷ 4 = 4 remainder 2, so it corresponds to the second position in the cycle, which is 4. Therefore, 2^18 ends in 4.

9^4 is calculated as 9 * 9 * 9 * 9, which is 81 * 81. 81 * 81 = 6561. The other options (6451, 8741, 9951) are incorrect products.

Any power of 5 (where the exponent is at least 1) ends in the digit 5 (5, 25, 125, 625...). Therefore, 5^25 must end in 5. Options like 0, 10, or 50 are impossible for powers of 5.

Any power of a number ending in 1 will always end in 1 (1*1=1). Therefore, 11 raised to any positive integer power, including 11, will result in a number with 1 as the last digit.

12^3 = 12 × 12 × 12 = 1728. This is a standard cube that students should memorize. First calculate 12 × 12 = 144, then 144 × 12 = 1728. Note that 1728 is also 12 dozen (a gross).

64 squared is 4096 (60^2 = 3600, 70^2 = 4900, so it must be between 60 and 70). 32 squared is 1024, and 128 squared is 16384. 54 squared is 2916.

The square root of 2 is a mathematical constant approximately equal to 1.41421. 1.735 is close to the square root of 3, while 1.448 and 1.659 are incorrect approximations.

The square root of 3 is approximately 1.73205. 1.414 is the square root of 2, and the other values provided (2.546, 1.221) are mathematically unrelated to the square root of 3.

Powers of 4 follow a cycle for the last digit: 4^1=4, 4^2=16, 4^3=64, 4^4=256. Odd powers end in 4 and even powers end in 6. Since 5 is odd, 4^5 ends in 4. (4^5 = 1024).