Number Operations and Properties

For any non-zero number n, n×n=n² and n²÷n=n. This works for all non-zero numbers including decimals (0.001, 0.16, 0.0016). However, 0×0=0, but 0÷0 is undefined (division by zero is impossible). Therefore, 0 is the exception.

This is a duplicate of question 72435. For any non-zero number n, n×n=n² and n²÷n=n. However, 0×0=0, but 0÷0 is undefined. Zero is the exception where the operation fails.

To solve this question, the user needs to know algebraic manipulation and simplification. The user must use the given equation to find the value of w and then substitute it into the expression (2w)2.

First, let's solve for w using the given equation:

13 = 13w/(1-w)

Multiplying both sides by (1-w), we get:

13 - 13w = 13w

Adding 13w to both sides, we get:

13 = 26w

Dividing both sides by 26, we get:

w = 1/2

Now that we have the value of w, we can substitute it into the expression (2w)2:

(2w)2 = (2(1/2))2 = 12

Thus, the correct answer is:

The Answer is: D. 1

17^8 = (17^4)^2 = 83521^2 = 6975757441. Option B correctly gives this value. The distractors (A, C, D) are close but incorrect values.

13^8 = (13^4)^2 = 28561^2 = 815730721. Option A correctly gives this value. The distractors are close but incorrect.

28^8 = (28^4)^2 = 614656^2 = 377801998336. Option D correctly gives this value. The other options are close but incorrect.

33^8 = (33^4)^2 = 1185921^2 = 1406408618241. Option B correctly gives this value. The distractors are close but incorrect.

13^7 = 13 × 13^6 = 13 × 4826809 = 62748517. Option B correctly gives this value. The other options are close but incorrect.

27^5 = 27 × 27^4 = 27 × 531441 = 14348907. Option C correctly gives this value. The distractors are close but incorrect.

The pattern is: a*b = (a*b) concatenated with (a*2). For 3*2: 3*2=6, 3*2=6, so 46. For 2*5: 2*5=10, 2*2=4, so 104. For 4*3: 4*3=12, 4*2=8, so 68. For 4*7: 4*7=28, 4*2=8, so 288, not 148. Wait, let me re-examine: 3*2=6 and 3*2=6 gives 46? That's strange. Let me try another interpretation: perhaps it's a*b = (a*b - a) concatenated with (a*b + a)? For 3*2: (6-3)=3, (6+3)=9, not 46. Let me try: a*b = (a*b/2) and (a*b)? No. The pattern might be: multiply a and b, then concatenate with (a*2). But 3*2=6, 3*2=6 gives 66, not 46. Let me reconsider: maybe the first digit is (a*b - 2)? For 3*2: 6-2=4, then 6 gives 46. For 2*5: 10-2=8, then 10 gives 810, not 104. This doesn't work. The actual pattern is likely: a*b = (a*b - a - b + something)... Let me just check: 4*7=28, and if we follow some pattern, maybe 148 is correct under some interpretation. However, based on standard logic, this pattern is unclear.

The operation is f(x) = 2x - 10. f(9) = 8. f(8) = 6. f(6) = 2. Applying the operation three times starting with 9 results in 2.

In standard arithmetic, the sum of 2 and 2 is 4. The other options represent common errors like concatenation (22) or decimal placement, but 4 is the only mathematically correct result.

Let the number be x. The problem states 2x - x/2 = 30. Multiplying by 2 to clear the fraction: 4x - x = 60, so 3x = 60. Solving for x gives x = 20. Distractors like 30 or 25 do not satisfy the equation.

Let the numbers be x and y, where x > y. Then x - y = 80 and x/y = 9, so x = 9y. Substituting: 9y - y = 80, so 8y = 80, giving y = 10. Then x = 9*10 = 90. The first number (x) is 90. Verification: 90 - 10 = 80 ✓ and 90/10 = 9 ✓.