Final answer:
The phrase 'four and nine hundred sixty-six ten-thousandths' as a decimal is 4.0966. This is understood by recognizing the whole number part and the fraction represented by 'ten-thousandths'. Dividing or multiplying by powers of 10 adjusts the decimal point accordingly, a key principle in understanding scientific notation.
Step-by-step explanation:
To write "four and nine hundred sixty-six ten-thousandths" as a decimal, we need to understand the value of each word in this phrase. 'Four' corresponds to the whole number part, so we start with a '4'. The phrase 'nine hundred sixty-six ten-thousandths' corresponds to the decimal part. Since 'ten-thousandths' is the same as division by 10,000 or multiplying by the power of 10 which in this case is 10 to the negative fourth power (10-4), it results in the decimal 0.0966. Therefore, combined we have 4.0966 as our final decimal.
Let's use a similar example to understand how dividing by powers of 10 affects our number. If we have 1.9436 and we divide by 10 (101), we get 0.19436. Further divisions, such as by 100 (102), result in 0.019436, and by 1000 we get 0.0019436. These divisions simply move the decimal point to the left for every power of ten.
The concept of scientific notation is also important here. It's a way to express very large or very small numbers more compactly. For instance, 965 becomes 9.65 x 102 because we moved the decimal two places to the left. Similarly, multiplying or dividing decimals by powers of 10 in scientific notation can greatly simplify calculations, especially in fields like chemistry.