Problem 105 - Algebra and Trigonometry

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$ \frac{0.00072 \times 0.003}{0.00024} $$

Step-by-Step Solution

Answer

The result of the given computations is \(9 \times 10^{-3}\).

Step 1: Convert decimal numbers to scientific notation

To perform the computations more easily, you can convert the decimal numbers to scientific notation first. \n \(0.00072 = 7.2 \times 10^{-4}\), \n \(0.003 = 3 \times 10^{-3}\), and \n \(0.00024 = 2.4 \times 10^{-4}\)

Step 2: Perform multiplication and division in scientific notation

Now you can perform the multiplication and division. Remember that you will multiply and divide as with simple numbers, but you will add the powers of 10 when multiplying and subtract them when dividing.\n \((7.2 \times 10^{-4}) \times (3 \times 10^{-3}) = 21.6 \times 10^{-7}\) \n Now, divide that by \(2.4 \times10^{-4}\) to get: \n \( \frac{21.6 \times 10^{-7}}{2.4 \times 10^{-4}} = 9 \times (10^{-7 +4}) = 9 \times 10^{-3}\)

Step 3: Round the decimal factor

The given question asks to round the decimal factor in the answer to two decimal places. But in this case, the decimal factor \(9\) has no decimal, so rounding is not necessary.

Key Concepts

Multiplication in Scientific Notation

When dealing with very large or very small numbers, scientific notation makes multiplication manageable. In scientific notation, a number is expressed as the product of a factor between 1 and 10 and a power of 10. For example, the number \(0.00072\) can be converted into \(7.2 \times 10^{-4}\). This method simplifies calculations enormously, especially when it comes to multiplying. To multiply numbers in scientific notation, first multiply the leading decimal factors. Using our earlier example, the multiplication of \(7.2\) and \(3\) from \(3 \times 10^{-3}\) gives \(21.6\). Next, add the exponents of the powers of ten. So, \((-4) + (-3) = -7\). Combine these two results to find the product in scientific notation:

• Decimal factor result: \(21.6\)
• Power of ten result: \(10^{-7}\)

Thus, the multiplication of these numbers yields \(21.6 \times 10^{-7}\). Be sure to check if your resulting decimal factor needs to be adjusted to stay within the range of 1 to 10, as this might involve altering the power of ten.

Division in Scientific Notation

Division in scientific notation follows a straightforward rule: divide the leading decimal factors, just as you do with regular numbers, and then subtract the exponents of the powers of ten. Let's use our exercise to introduce this concept:Suppose we have the numerator \(21.6 \times 10^{-7}\) and the denominator \(2.4 \times 10^{-4}\). Start by performing the division of the decimal factors. Divide \(21.6\) by \(2.4\) to get \(9\).Next, work with the exponents of ten: subtract the exponent of the denominator from the exponent of the numerator. That means \(-7 - (-4) = -7 + 4 = -3\).Combine these two results:

• Decimal factor: \(9\)
• Power of ten: \(10^{-3}\)

This division results in \(9 \times 10^{-3}\). It's essential to ensure the decimal factor fits within the 1 to 10 range, even in division.

Rounding Decimal Factor

Rounding is a crucial step when working with scientific notation, especially when it's required to present the answer in a specified number of decimal places. The rule is simple: check the decimal part of the factor and round it appropriately.For our exercise, if we reached a solution like \(9.174 \times 10^{-3}\), and were required to round to two decimal places, we'd round \(9.174\) to \(9.17\). However, since our result was \(9\), no rounding was necessary because the decimal factor was already a whole number.The method for rounding is the same as for any typical decimal number:

• Check the digit after the last required decimal place.
• If it is 5 or greater, round up.
• If it is less than 5, round down.

This approach keeps scientific notation answers neat and consistent, which is particularly helpful for ensuring clarity in scientific and mathematical communications.