This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to your math problems: 1. A rectangular plot of land has an area of 14 m^2. If the width is (4 - sqrt(2)) m, find the length of the land. Step 1: Write the formula for the area of a rectangle and substitute the given values. The area of a rectangle is given by A = L × W, where A is the area, L is the length, and W is the width. We are given A = 14 m^2 and W = (4 - sqrt(2)) m. To find the length L, we rearrange the formula: L = (A)/(W) L = (14)/(4 - sqrt(2)) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of (4 - sqrt(2)) is (4 + sqrt(2)). L = (14)/(4 - sqrt(2)) × 4 + sqrt(2)4 + sqrt(2) L = 14(4 + sqrt(2))(4 - sqrt(2))(4 + sqrt(2)) Step 3: Expand the numerator and denominator and simplify. Using the difference of squares formula (a-b)(a+b) = a^2 - b^2 for the denominator: L = 14(4 + sqrt(2))4^2 - (sqrt(2))^2 L = 14(4 + sqrt(2))16 - 2 L = 14(4 + sqrt(2))14 L = 4 + sqrt(2) The length of the land is (4 + sqrt(2)) m. 2. Simplify _6^12 + _6^3 Step 1: Apply the logarithm addition property. The property states that _b x + _b y = _b (x × y). Here, b=6, x=12, and y=3. _6^12 + _6^3 = _6(12 × 3) Step 2: Calculate the product inside the logarithm. _6(12 × 3) = _6(36) Step 3: Evaluate the logarithm. We need to find the power to which 6 must be raised to get 36. Since 6^2 = 36, then _6(36) = 2. _6^12 + _6^3 = 2 Drop the next question.