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
Step 1: Rewrite the expression using logarithm properties. The expression to evaluate is _2 8 sqrt((a^5 b)/(c^3)). Using the product rule of logarithms, _x (MN) = _x M + _x N: _2 8 + _2 sqrt((a^5 b)/(c^3)) Rewrite the square root as a power: sqrt(X) = X^(1)/(2). _2 8 + _2 ((a^5 b)/(c^3))^(1)/(2) Step 2: Apply the power rule of logarithms. Using the power rule, _x M^P = P _x M: _2 8 + (1)/(2) _2 ((a^5 b)/(c^3)) Step 3: Apply the quotient and product rules of logarithms inside the parenthesis. Using the quotient rule, _x ((M)/(N)) = _x M - _x N: _2 8 + (1)/(2) (_2 (a^5 b) - _2 (c^3)) Using the product rule, _x (MN) = _x M + _x N: _2 8 + (1)/(2) (_2 a^5 + _2 b - _2 c^3) Step 4: Apply the power rule again for a^5 and c^3. _2 8 + (1)/(2) (5 _2 a + _2 b - 3 _2 c) Step 5: Evaluate _2 8. Since 2^3 = 8, we have _2 8 = 3. Step 6: Substitute the given values into the expression. Given: _2 a = 3 _2 b = -2 _2 c = 5 Substitute these values into the expression from Step 4: 3 + (1)/(2) (5(3) + (-2) - 3(5)) Step 7: Perform the arithmetic calculations. 3 + (1)/(2) (15 - 2 - 15) 3 + (1)/(2) (-2) 3 - 1 2 The final answer is 2.