If $x, y, z$ and $w$ be the digits of a number beginning from the left, the number is

### Concept & Formula The algebraic representation of a number relies on the standard base-10 place value system. Each digit is multiplied by a power of 10 corresponding to its position. ### Step-by-Step Solution **Given:** * The digits of a four-digit number, reading from left to right, are $x$, $y$, $z$, and $w$. **Calculation / Deduction:** In a standard base-10 number system, the positions from right to left are units ($10^0$), tens ($10^1$), hundreds ($10^2$), and thousands ($10^3$). Because the sequence $x, y, z, w$ begins from the left, the positions are assigned as follows: * $x$ is in the thousands place. * $y$ is in the hundreds place. * $z$ is in the tens place. * $w$ is in the units (ones) place. To find the mathematical value of the number, multiply each digit by its respective place value and sum them up: $$\text{Value} = (x \times 10^3) + (y \times 10^2) + (z \times 10^1) + (w \times 10^0)$$ $$\text{Value} = 10^3x + 10^2y + 10z + w$$ ### Exam Strategy & Shortcut When translating digits into algebraic values from left to right, instantly assign the highest power of 10 to the first variable. Since it's a 4-digit number, the first power is $10^3$ assigned to $x$. Option (d) perfectly matches this descending power structure. ### Common Pitfall The most common mistake is choosing option (a) $xyzw$. In arithmetic, placing digits next to each other forms a number (like $4567$), but in algebra, placing variables side-by-side indicates multiplication ($x \times y \times z \times w$). Option (c) is also a trap where the powers of 10 are reversed. ### Final Answer Therefore, the correct answer is **$10^3x + 10^2y + 10z + w$**.