A number is written as 25 to number base x. twice the number is written as 52 to base x. find the value of x.

1. Find the value of x. Step 1: Convert the numbers from base x to base 10. The number written as 25_x in base x is equivalent to 2 × x^1 + 5 × x^0 = 2x + 5 in base 10. Twice the number is written as 52_x in base x, which is equivalent to 5 × x^1 + 2 × x^0 = 5x + 2 in base 10. Step 2: Set up the equation based on the problem statement. "Twice the number is written as 52_x" means 2 × (2x + 5) = 5x + 2. Step 3: Solve the equation for x. 4x + 10 = 5x + 2 Subtract 4x from both sides: 10 = x + 2 Subtract 2 from both sides: x = 8 The value of x must be greater than the digits used (5 and 2), so x=8 is a valid base. The value of x is 8 2. Evaluate the expressions to four significant figures. Given sqrt(5) = 2.2361 and sqrt(7) = 2.6458. a) (i). sqrt(72) - (3)/(sqrt(7)) Step 1: Simplify the terms. We simplify sqrt(72) as sqrt(36 × 2) = 6sqrt(2). We rationalize the denominator for (3)/(sqrt(7)): (3)/(sqrt(7)) = 3sqrt(7)7. The expression becomes 6sqrt(2) - 3sqrt(7)7. Assumption: Since sqrt(2) is not provided, but the problem asks for evaluation without tables/calculator using given values, we assume sqrt(2) ≈ 1.4142 (a commonly known approximation to similar precision as given values) is expected. Step 2: Substitute the given and assumed values. 6sqrt(2) - 3sqrt(7)7 ≈ 6 × 1.4142 - (3 × 2.6458)/(7) ≈ 8.4852 - (7.9374)/(7) ≈ 8.4852 - 1.133914... Step 3: Perform the subtraction and round to four significant figures. ≈ 7.351285... Rounding to four significant figures, we get 7.351 b) (ii). (4)/(sqrt(5)) + 2sqrt(5) Step 1: Simplify the expression by rationalizing the denominator. (4)/(sqrt(5)) + 2sqrt(5) = 4sqrt(5)5 + 2sqrt(5) Step 2: Combine the terms. = ((4)/(5) + 2)sqrt(5) = ((4)/(5) + (10)/(5))sqrt(5) = (14)/(5)sqrt(5) = 2.8sqrt(5) Step 3: Substitute the given value for sqrt(5) and round to four significant figures. = 2.8 × 2.2361 = 6.26108 Rounding to four significant figures, we get 6.261 3. Evaluate _2 sqrt(((a^5 b)/(c^2))) Given _2 a = 3, _2 b = -2, and _2 c = 5. Step 1: Rewrite the expression using logarithm properties. _2 sqrt(((a^5 b)/(c^2))) = _2 ((a^5 b)/(c^2))^1/2 Using the power rule _k M^p = p _k M: = (1)/(2) _2 ((a^5 b)/(c^2)) Step 2: Apply the quotient and product rules for logarithms. Using the quotient rule _k (M)/(N) = _k M - _k N: = (1)/(2) (_2 (a^5 b) - _2 (c^2)) Using the product rule _k (MN) = _k M + _k N: = (1)/(2) ((_2 a^5 + _2 b) - _2 c^2) Step 3: Apply the power rule again. = (1)/(2) (5_2 a + _2 b - 2_2 c) Step 4: Substitute the given values. = (1)/(2) (5(3) + (-2) - 2(5)) = (1)/(2) (15 - 2 - 10) = (1)/(2) (3) = 1.5 The value is 1.5 4. Find the mean of the new set of numbers. Given that the mean of x_1, x_2, x_3, x_4 is 7. Step 1: Use the definition of the mean to find the sum of the original numbers. (x_1 + x_2 + x_3 + x_4)/(4) = 7 x_1 + x_2 + x_3 + x_4 = 7 × 4 = 28 Step 2: Set up the expression for the mean of the new set of numbers. The new numbers are 5 + x_1, 5 + x_2, 5 + x_3, 5 + x_4. Mean = ((5 + x_1) + (5 + x_2) + (5 + x_3) + (5 + x_4))/(4) Step 3: Simplify and substitute the sum of the original numbers. Mean = (4 × 5 + (x_1 + x_2 + x_3 + x_4))/(4) Mean = (20 + 28)/(4) Mean = (48)/(4) Mean = 12 The mean of the new numbers is 12 5. Calculate the mean age of the class. The frequency table is: Age (years) (x) | 12 | 13 | 14 | 15 Frequency (f) | 8 | 10 | 4 | 2 Step 1: Calculate the sum of frequencies ( f). f = 8 + 10 + 4 + 2 = 24 Step 2: Calculate the sum of (age × frequency) ( fx). fx = (12 × 8) + (13 × 10) + (14 × 4) + (15 × 2) fx = 96 + 130 + 56 + 30 fx = 312 Step 3: Calculate the mean age using the formula Mean = ( fx)/( f). Mean = (312)/(24) Mean = 13 The mean age of the class is 13 years Drop the next question! 📸