Scientific Notation - IB Mathematics Analysis and Approaches

If $x = a \times 10^n$ and $y = b \times 10^{n+2}$ , where $1 \le a, b < 10$ and $n$ is a positive integer, which of the following best approximates the value of $x+y$ ?
Answer: $b \times 10^{n+2}$ To add $x$ and $y$ , we must have the same exponent. We can write $x = a \times 10^n = (a \times 10^{-2}) \times 10^{n+2} = 0.0a \times 10^{n+2}$ . The sum is $x+y = (0.0a + b) \times 10^{n+2}$ . Since $1 \le b < 10$ and $0.01 \le 0.0a < 0.1$ , the term $0.0a$ is much smaller than $b$ . Therefore, the sum is dominated by $y$ , and $x+y \approx y = b \times 10^{n+2}$ .