Problem 69 - Elementary Algebra

Question

Convert the following problems from scientific form to standar $$ 6.009 \times 10^{7} $$

Step-by-Step Solution

Answer

Question: Convert the number \(6.009 \times 10^{7}\) from scientific notation to standard notation. Answer: 60,090,000

Step 1: Identify the exponent

The exponent of 10 in the given scientific notation is 7, which means we need to move the decimal point 7 places to the right.

Step 2: Move the decimal point

Move the decimal point 7 places to the right, i.e. $$6.009 \times 10^7 = 60090000$$.

Step 3: Write the answer in standard notation

After moving the decimal point, we get the number in standard notation as: $$60,090,000$$

Key Concepts

Scientific Notation

Understanding scientific notation is crucial for working with very large or very small numbers efficiently. It's a way of expressing numbers as a product of two parts: a decimal number and an exponent of 10. Typically, the decimal is between 1 and 10, and the exponent shows how many places the decimal point would move to convert it into a standard number. For example, in scientific notation, the number \( 6.009 \times 10^{7} \) signifies that the decimal point in the decimal 6.009 must be moved 7 places.
Scientific notation is not just a mathematician's shortcut; it is widely used in fields like science and engineering to make computation with enormously large or tiny numbers more manageable.

Standard Notation

In contrast to scientific notation, standard notation is the normal way of writing numbers without exponents. It's the number format we use in daily life, from reading prices to counting items. Converting from scientific notation to standard notation involves expanding the number to show its full value as we would usually write it. For instance, taking \( 6.009 \times 10^{7} \) and converting it to 60,090,000 makes it understandable in the context of ordinary arithmetic and allows us to perceive the scale of the number at a glance.

Exponents

Exponents play a key role in scientific notation by indicating how many times to multiply the base, which is always 10, by itself. In our example, the number 7 is the exponent in \( 10^{7} \), telling us that we need to multiply 10 by itself 7 times. However, in terms of moving the decimal place, it tells us the number of places the decimal must shift to convert it into a standard notation number. A positive exponent means we move the decimal to the right, creating a larger number, while a negative exponent would mean moving the decimal to the left, creating a smaller number.

Positive Exponent Example:

• \( 3 \times 10^{2} \) would become 300, because the decimal moves two places to the right.

Negative Exponent Example:

• \( 4.5 \times 10^{-3} \) would become 0.0045, because the decimal moves three places to the left.

Decimal Point Movement

The movement of the decimal point is a visual representation of the mathematical process involved in changing between scientific and standard notation. To convert scientific notation to standard form, you look at the exponent on the 10, which indicates the number of places to move the decimal point. Moving the decimal point to the right makes the number larger, as seen with positive exponents, and moving it to the left makes the number smaller, as encountered with negative exponents.
In our exercise with \( 6.009 \times 10^{7} \), moving the decimal 7 places to the right gives us the number in standard form, which is 60,090,000. This process is straightforward but requires careful attention to the position of the decimal and the magnitude of the exponent.