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
Hey I — ready when you are. I have carefully reviewed the image again, including all the cropped versions. The expression for question 15, "Simplify by rationalising the denominator:", is still not visible. There is a blank space where the fraction should be. It seems the expression itself is missing from the question paper. Since I cannot solve question 15 without the expression, I will solve question 12 for you, which is fully visible. 12. Use logarithm tables to evaluate: ((36.72 × 0.46)/(185.4^2))^(1)/(2) - (1)/(3) Step 1: Simplify the exponent. (1)/(2) - (1)/(3) = (3)/(6) - (2)/(6) = (1)/(6) So the expression becomes: N = ((36.72 × 0.46)/(185.4^2))^(1)/(6) Step 2: Take the logarithm of both sides. N = [ ((36.72 × 0.46)/(185.4^2))^(1)/(6) ] Using logarithm properties, (a^b) = b a and ((a × b)/(c)) = a + b - c: N = (1)/(6) [ (36.72) + (0.46) - (185.4^2) ] N = (1)/(6) [ (36.72) + (0.46) - 2 (185.4) ] Step 3: Find the logarithms of the numbers using logarithm tables (or a calculator for precision, as tables are not provided). (36.72): Characteristic = 1 (since 36.72 is between 10 and 100). Mantissa for 3672 is approximately 0.5649. So, (36.72) ≈ 1.5649. (0.46): Characteristic = 1 (since 0.46 is between 0.1 and 1). Mantissa for 4600 is approximately 0.6628. So, (0.46) ≈ 1.6628. (185.4): Characteristic = 2 (since 185.4 is between 100 and 1000). Mantissa for 1854 is approximately 0.2681. So, (185.4) ≈ 2.2681. Step 4: Substitute the logarithm values into the equation for N. N = (1)/(6) [ 1.5649 + (1.6628) - 2(2.2681) ] N = (1)/(6) [ 1.5649 + (-1 + 0.6628) - 4.5362 ] N = (1)/(6) [ 1.5649 - 0.3372 - 4.5362 ] N = (1)/(6) [ 1.2277 - 4.5362 ] N = (1)/(6) [ -3.3085 ] N = -0.551416... Step 5: Convert the negative logarithm to the standard form with a negative characteristic and positive mantissa. N = -1 + 0.448583... N ≈ 1.4486 Step 6: Find the antilogarithm of 1.4486. Antilog of 0.4486 is approximately 2.810. Since the characteristic is 1, the decimal point is placed such that there is one zero before the first non-zero digit. N ≈ 0.2810 Therefore, the value is 0.2810. If you'd like to try another question from the image, just let me know which one! 📸