Problem 36 - Elementary Algebra

Question

Convert the numbers used in the following problems to scientific notation. The mass of an amoeba is about 0.000004 gram.

Step-by-Step Solution

Answer

Question: Convert the mass of an amoeba, which is 0.000004 gram, to scientific notation. Answer: The mass of the amoeba in scientific notation is 4 x 10^{-6} gram.

Step 1: Identify the Number of Decimal Places to Move

Since the mass of the amoeba is 0.000004 gram, we need to move the decimal point 6 places to the right to get a number between 1 and 10 (4).

Step 2: Rewrite the Mass as the Product of the New Number and a Power of 10

Now that we have moved the decimal point 6 places to the right, we can rewrite the mass as the product of 4 and a negative power of 10. Since we moved the decimal point to the right, the exponent of 10 will be negative. In this case, the exponent will be -6.

Step 3: Write the Mass in Scientific Notation

Having identified the number between 1 and 10 (4) and the correct power of 10 (-6), we can now write the mass of the amoeba in scientific notation: 4 x 10^{-6} gram.

Key Concepts

Decimal Places

When converting a number into scientific notation, one of the first steps is to decide how many decimal places you need to move in order to have a number between 1 and 10. This makes the number manageable and easier to work with in scientific notation.

In our example, the amoeba's mass is 0.000004 grams. Notice how small this number is. We need to move the decimal point to the right until we reach the first non-zero digit. In this case, we move the decimal point 6 places to the right until we reach 4. This means we will express the number as 4 times a power of ten. The trick here is to ensure that the resulting number is always between 1 and 10, which is a key characteristic of scientific notation.

Powers of 10

Once you've determined how many decimal places to move, the next step is to express this shift in terms of a power of 10. The power of 10 represents how many times you have multiplied or divided by 10 to change the number into its new form.

In scientific notation, moving the decimal to the right corresponds to multiplying the number by a negative power of 10. Each place you move to the right increases the negative exponent by one. For instance, for the amoeba's mass, we moved the decimal 6 places to the right, so the power of 10 is \(-6\). This gives us the expression: 4 multiplied by \(10^{-6}\).

The power of 10 simplifies the notation, and makes it apparent how large or small a number is, by clearly indicating the order of magnitude.

Negative Exponent

Negative exponents in the context of scientific notation indicate the number of times we divide by 10, rather than multiply. This is crucial when dealing with very small numbers. Instead of writing a tiny number with lots of zeros, a negative exponent in the power of 10 skillfully compresses that information into a compact form.

For instance, using a negative exponent in our example, \(4 \times 10^{-6}\), efficiently tells us that we have to divide 4 by \(10^6\) or a million, to reach the initial tiny mass of 0.000004 grams. This approach not only makes the numbers shorter but also conveys more intuitive information about the scale and size of the number, showing us how far we had to move the decimal place to get the decimal-point-free number we started with.