If 251m = 100two. Find the value of m

You're on a roll — Here are the solutions to the questions from the image: 1a. If 251_m = 100_two. Find the value of m Step 1: Convert both numbers to base 10. For 251_m: 2 × m^2 + 5 × m^1 + 1 × m^0 = 2m^2 + 5m + 1 For 100_two: 1 × 2^2 + 0 × 2^1 + 0 × 2^0 = 4 + 0 + 0 = 4 Step 2: Set the base 10 expressions equal to each other. 2m^2 + 5m + 1 = 4 Step 3: Rearrange into a quadratic equation. 2m^2 + 5m - 3 = 0 Step 4: Solve the quadratic equation for m. We can use factoring. Find two numbers that multiply to 2 × (-3) = -6 and add to 5. These numbers are 6 and -1. 2m^2 + 6m - m - 3 = 0 2m(m + 3) - 1(m + 3) = 0 (2m - 1)(m + 3) = 0 This gives two possible values for m: 2m - 1 = 0 2m = 1 m = (1)/(2) m + 3 = 0 m = -3 Step 5: Determine the valid base. A number base must be a positive integer greater than the largest digit used in the number. In 251_m, the largest digit is 5, so m must be greater than 5. Neither (1)/(2) nor -3 satisfy this condition. However, in some contexts, m is simply a variable in an equation, and the base interpretation is secondary. Given the context of "Find the value of m", we provide the mathematical solutions. If this were strictly a number base problem, there would be no valid integer solution for m > 5. Assuming it's a general algebraic problem: m = (1)/(2) or m = -3 1b. Simplify the following; (0.00047 × 0.0093)/(0.00000065). Correct to 3 d.p. Step 1: Perform the multiplication in the numerator. 0.00047 × 0.0093 = 0.000004371 Step 2: Divide the result by the denominator. (0.000004371)/(0.00000065) To simplify, multiply numerator and denominator by 10^8 to remove decimals: (0.000004371 × 10^8)/(0.00000065 × 10^8) = (437.1)/(65) Step 3: Perform the division. (437.1)/(65) ≈ 6.724615... Step 4: Round to 3 decimal places. The fourth decimal place is 6, so we round up the third decimal place. 6.725 2a. 36^1/2 × 64^1/2 × 5^0 Step 1: Evaluate each term. 36^1/2 = sqrt(36) = 6 64^1/2 = sqrt(64) = 8 5^0 = 1 (Any non-zero number raised to the power of 0 is 1) Step 2: Multiply the results. 6 × 8 × 1 = 48 2b. 4^-1/2 × 16^3/4 × 32^-2/5 Step 1: Evaluate each term. 4^-1/2 = (1)/(4^1/2) = (1)/(sqrt(4)) = (1)/(2) 16^3/4 = ([4]16)^3 = (2)^3 = 8 32^-2/5 = (1)/(32^2/5) = (1)/(([5]32))^2 = (1)/((2)^2) = (1)/(4) Step 2: Multiply the results. (1)/(2) × 8 × (1)/(4) = (8)/(8) = 1 3a. 5(sqrt(1.6) × sqrt(0.4)) Step 1: Combine the square roots. 5(sqrt(1.6 × 0.4)) Step 2: Perform the multiplication inside the square root. 1.6 × 0.4 = 0.64 Step 3: Evaluate the square root. sqrt(0.64) = 0.8 Step 4: Multiply by 5. 5 × 0.8 = 4 3b. 2sqrt(6) - sqrt(3)3 Step 1: Check if the expression can be simplified. The terms in the numerator, 2sqrt(6) and sqrt(3), are not like terms, so they cannot be combined. The denominator is 3. We can write this as two separate fractions if desired, but it doesn't simplify further. 2sqrt(6)3 - sqrt(3)3 Since no further simplification is possible, the expression remains as is. 2sqrt(6) - sqrt(3)3 4. With the aid of diagram explain the following; i) Set (ii) Singleton Set (iii) Venn diagram i) Set A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. Sets are typically denoted by capital letters, and their elements are listed within curly braces. Example: The set of even numbers less than 10 is A = \2, 4, 6, 8\. Diagram: ` A +---+ | 2 | | 4 | | 6 | | 8 | +---+ ` ii) Singleton Set A singleton set (or unit set) is a set containing exactly one element. Example: The set of even prime numbers is B = \2\. Diagram: ` B +---+ | 2 | +---+ ` iii) Venn diagram A Venn diagram is a visual representation used to show the relationships between different sets. It uses overlapping circles or other shapes to illustrate the logical relationships between sets, with the universal set often represented by a rectangle. Example: Let U = \1, 2, 3, 4, 5, 6\, P = \1, 2, 3\, and Q = \3, 4, 5\. Diagram: ` +---------------------------------+ | U | | +-------+ +-------+ | | | P | | Q | | | | 1 2 |-----| 4 5 | | | | | 3 | | | | +-------+ +-------+ | | | +---------------------------------+ ` In this diagram, the rectangle represents the universal set U. Circle P contains elements 1, 2, 3. Circle Q contains elements 3, 4, 5. The overlap shows element 3, which is common to both sets P and Q. Element 6 is in U but not in P or Q. Last free one today — make it count tomorrow, or type /upgrade for unlimited.