answer:Let x = sqrt{12 + !sqrt{12 + !sqrt{12 + !sqrt{12 + cdots}}}}. Then we have x = sqrt{12 + x}. Squaring both sides, we get x^2 = 12 + x. Rearranging, we have x^2 - x - 12 = 0. Factoring, we have (x-4)(x+3) = 0. Since x represents a positive length, we have x = 4. Therefore, sqrt{12 +!sqrt{12 + !sqrt{12 + !sqrt{12 + cdots}}}} = boxed{4}. The answer is: 4

answer:To solve this problem, we need to determine the value of x, which represents the largest prime number less than or equal to 61. Let's break down the information given: Number of terms in the sequence: 58 Product of all prime numbers less than or equal to 61: P Number of primes appearing in the sequence: N We can set up the equation as follows: N = 58 To find the value of x, we need to find the largest prime number less than or equal to 61. By checking the prime numbers less than or equal to 61, we can see that the largest prime number is 61 itself. Therefore, the value of x is 61. The answer is: 61

answer:Using the properties of logarithms, we can rewrite the equation as log_{10x^2} (10 cdot 100x^3) = -2. Simplifying, we get log_{10x^2} (1000x^3) = -2. Expressing everything in terms of powers of 10, we have (10x^2)^{-2} = (10^3 cdot x^3). Simplifying further, we get (10x^2)^{-2} = 10^3 cdot x^3. Taking the reciprocal of both sides, we have (10x^2)^{2} = frac{1}{10^3} cdot frac{1}{x^3}. Simplifying, we get 10^4x^4 = frac{1}{10^3} cdot frac{1}{x^3}. Rearranging, we have x^{12} = frac{1}{10^7}. Taking the reciprocal, we get frac{1}{x^{12}} = 10^7. Therefore, the decimal representation of frac{1}{x^{12}} is boxed{10000000}.The answer is: 10000000

answer:To solve this problem, we need to determine the value of x, which represents the number of calories James burns per minute during his run. Let's break down the information given: Number of bags of Cheezits: 3 Weight per bag of Cheezits: 2 ounces Calories per ounce of Cheezits: 150 Total calories consumed from Cheezits: 3 * 2 * 150 = 900 Run duration: 40 minutes Excess calories consumed: 420 Calories burned per minute: x We can set up the equation as follows: Total calories consumed - (Calories burned per minute * Run duration) = Excess calories consumed 900 - (x * 40) = 420 Let's simplify and solve for x: 900 - 40x = 420 To isolate x, we subtract 900 from both sides of the equation: 900 - 900 - 40x = 420 - 900 -40x = -480 To solve for x, we divide both sides of the equation by -40: x = -480 / -40 x = 12 The value of x is 12. 12 The answer is: 12