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
To find the value of x, we will simplify the given equation using the laws of indices. Step 1: Simplify the right-hand side of the equation and express all bases on the left-hand side as powers of 2. We know that any non-zero number raised to the power of 0 is 1, so 10^0 = 1. Also, 8 = 2^3, 16 = 2^4, and 4 = 2^2. Substitute these into the equation: ((2^3)^x)/(2) × (2^4)^(x)/(2)2^x × (2^2)^x = 1 Step 2: Apply the power rule (a^m)^n = a^mn to simplify the exponents. (2^3 × x)/(2) × 2^4 × (x)/(2)2^x × 2^2 × x = 1 (2^3x)/(2) × 2^2x2^x × 2^2x = 1 Step 3: Apply the product rule a^m × a^n = a^m+n to combine terms in the numerator and denominator. Numerator: 2^(3x)/(2) × 2^2x = 2^(3x)/(2) + (4x)/(2) = 2^(7x)/(2) Denominator: 2^x × 2^2x = 2^x + 2x = 2^3x The equation becomes: (2^7x)/(2)2^3x = 1 Step 4: Apply the quotient rule (a^m)/(a^n) = a^m-n to simplify the left-hand side. 2^(7x)/(2) - 3x = 1 Simplify the exponent: (7x)/(2) - 3x = (7x)/(2) - (6x)/(2) = (x)/(2) So the equation is: 2^(x)/(2) = 1 Step 5: Solve for x. Since 2^0 = 1, we can equate the exponents: (x)/(2) = 0 x = 0 × 2 x = 0 What's next?