Question

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 654,000 \cdot 0.000042 $$

Short answer

The result of the multiplication in scientific notation is \( 2.7468 \times 10^{-2} \) and in decimal form is 0.027468

Step-by-step solution

Perform the multiplication

Using a calculator, perform the multiplication \(654,000 \cdot 0.000042.\) This should give you a decimal value.

Write the result in scientific notation

Convert the decimal result obtained from the multiplication into scientific notation. This involves moving the decimal point to the right or left until there is only one non-zero digit to its left. The number of places the point was moved will be the exponent on 10. If the point was moved to the left, the exponent will be positive and if the decimal was moved to the right, the exponent will be negative.

Convert the scientific notation into decimal form

Now convert the scientific notation result into its decimal form. If the exponent on 10 in the scientific notation is positive, move the decimal point to the right by the number of places equal to the exponent, filling in any gaps with zeros. If the exponent is negative, move the decimal point to the left by the number of places equal to the exponent, filling in any new gaps with zeros.

Key concepts

Calculator Usage

In today's digital age, calculators are an essential tool not only for complex calculations but even for basic arithmetic that involves large numbers or small decimals. When using a calculator for multiplication, such as evaluating the expression 654,000 × 0.000042, start by entering the first number precisely as it is. Next, use the multiplication function (typically denoted by an 'x' or '*' symbol), followed by the second number.
It's important to understand your calculator's layout and functions. Some calculators follow different operations for inputting and displaying very large or small numbers, which is where understanding scientific notation becomes handy. Keep an eye on how your calculator handles overflow, i.e., numbers too large for the display, as well as underflow, numbers too small that might be displayed in scientific notation automatically.
In the context of our exercise, the result displayed should be the product of the two given numbers. If you find the result is given in scientific notation, it's fine, as we need that anyway for the next step of our task. If not, you might need to convert it.

Multiplying Decimals

When it comes to multiplying decimals, precision is key. Multiplication of decimals is not much different from multiplication of whole numbers: you multiply the numbers disregarding the decimal points and then add the decimal point in the correct place in the resulting number. The important rule is that the position of the decimal in the product is determined by the combined number of decimal places in the factors being multiplied.
For example, if you multiply 2.5 (one decimal place) by 0.002 (three decimal places), the result should have a total of four decimal places. This can be tricky when working with large numbers or with numbers with a lot of decimal places, which is why a calculator is a handy tool for such tasks. After the calculation, the result can often be a number with many zeroes, which leads us to the necessity of converting it into a more readable form: scientific notation.

Converting Decimals to Scientific Notation

Converting decimals to scientific notation is a process that makes very large or small numbers more manageable. Scientific notation expresses numbers as a product of two factors: a decimal with only one non-zero digit to the left of the decimal point, and a power of ten.
To convert a decimal to scientific notation, identify the most significant non-zero digit. Move the decimal point so that it's just to the right of that digit. Count how many places you've moved it: that's your exponent for the base 10. If you've moved the decimal point to the left, your exponent is positive; if to the right, it's negative. For instance, converting 0.000042 into scientific notation results in 4.2 × 10^{-5} because we move the decimal point five places to the right. In our original exercise, after multiplying the decimals, we would follow these steps to express the answer in scientific notation, thereby simplifying the comprehension and communication of very large or very small numbers.