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3 Responses
Pretty sure it’s 3.7 × 10^-4, because in standard form the first number must be between 1 and 10; 0.37 × 10^-3 is the same value but not in standard form. I do get tangled with the minus sign too, but moving the decimal four places right means 10^-4.
Great question. In standard form we write a number as a x 10^n with 1 ≤ a < 10. For decimals smaller than 1, the exponent is negative. For 0.00037, I look for the first number between 1 and 10: slide the decimal four places to the left to make 3.7, so n = -4 and the standard form is 3.7 x 10^-4. The expression 0.37 x 10^-3 is equal to the same value (since 0.37 x 0.001 = 0.00037), but it isn’t in standard form because 0.37 is less than 1 and doesn’t meet the 1 ≤ a < 10 rule. As a quick worked example, take 0.0052. I move the decimal to the left until I get a number between 1 and 10: that takes three places to reach 5.2, so I write 5.2 x 10^-3. If I wrote 0.52 x 10^-2, it would still be the same number, just not in standard form because the leading factor is below 1. I like to check by multiplying back: 5.2 x 10^-3 = 5.2 x 0.001 = 0.0052, so the count of three places (and the negative exponent) matches.
I always remind myself: in standard form the first factor must be between 1 and 10, and for small numbers you move the decimal to the right and use a negative exponent equal to the number of moves. Example: 0.00037 → move 4 places to get 3.7, so 0.00037 = 3.7 × 10^-4 (not 0.37 × 10^-3, since 0.37 isn’t between 1 and 10).