Problem 21 - Algebra 1: Concepts and Skills

Question

Write the number in decimal form. $$ 8 \times 10^{3} $$

Step-by-Step Solution

Answer

8000

Step 1: Identify the base and the exponent

In the expression \(8 \times 10^{3}\), 10 is the base and 3 is the exponent. The coefficient here is 8.

Step 2: Shift the decimal

Since the exponent is positive, we shift the decimal point in the coefficient 3 places to the right. As the coefficient is 8 (which is whole number, the decimal place is originally at the rightmost position, just after 8)

Step 3: Writing the number in decimal form

Shifting the decimal in the coefficient 8 for three places to the right, we get 8000 as the decimal form of the given scientific notation.

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. This notation comprises two parts: a coefficient and a base raised to an exponent. In the context of our exercise, the given number is in scientific notation and is written as \(8 \times 10^{3}\).

Here's a tip: whenever you face scientific notation, remember that the base will almost always be 10, and your main task is to place the decimal point correctly in relation to the coefficient, based on the exponent.

Exponent

An exponent, found in scientific notation, is a small number written slightly above and to the right of a base number. It tells you how many times to multiply the base number by itself. In our example \(10^{3}\), the exponent is 3. This means that we multiply 10 by itself 3 times: \(10 \times 10 \times 10\), which equals 1,000.

Key Insight

When converting from scientific to decimal notation, if the exponent is positive, as in our example, you move the decimal point in the coefficient to the right as many times as the exponent indicates.

Base in Mathematics

In any mathematical expression, the base is the number to be multiplied by itself a number of times as indicated by the exponent. For most scientific notations, the base used is 10. The choice of base 10 is not arbitrary—it's chosen for convenience since our numeral system is decimal (base 10), and it simplifies calculations and conversions.

So, why is understanding the base important? When using scientific notation, comprehending the base allows you to easily manipulate numbers on both a small and large scale, essential for fields like science and engineering.

Coefficient

The coefficient in scientific notation is the number that is multiplied by the base raised to the exponent. It is typically a number equal to or greater than 1 and less than 10, but in some cases, like the one presented in the exercise, it can be a whole number like 8. The coefficient gives the significant digits of the number, carrying the most impactful figures of the value.

Remember, when you're converting from scientific notation to decimal form, the coefficient determines where the decimal point starts. In \(8 \times 10^{3}\), the coefficient 8 dictates that your starting point is before shifting to match the exponent's direction.