Question

to compare two numbers that are in scientific notation, first compare the -------. if they are the same, then compare the -------.
options:
first number, next number

exponents; decimals

decimals; exponents

Explanation:

To compare two numbers in scientific notation (in the form \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer), we follow these steps:

Step 1: Compare the exponents

The exponent (\( n \)) in scientific notation represents the order of magnitude. A larger exponent means a larger number (if the coefficient \( a \) is positive) because it indicates a greater power of 10. For example, \( 3 \times 10^5 \) is larger than \( 2 \times 10^4 \) because \( 5 > 4 \).

Step 2: If exponents are equal, compare the coefficients (decimals)

If the exponents (\( n \)) are the same, we then compare the coefficient (\( a \), the decimal part) directly. The larger coefficient will correspond to the larger number. For example, \( 5 \times 10^3 \) is larger than \( 3 \times 10^3 \) because \( 5 > 3 \).

Answer:

To compare two numbers in scientific notation, first compare the \(\boldsymbol{\text{exponents}}\). If they are the same, then compare the \(\boldsymbol{\text{decimals}}\) (the coefficients).

So the correct option (matching the red box: "decimals; exponents" is incorrect order, the blue box: "exponents; decimals" is correct order) is the blue box: exponents; decimals