Problem 47 - College Algebra with Modeling and Visualization

Question

Write the number in standard form. $$ 5 \times 10^{5} $$

Step-by-Step Solution

Answer

The standard form is 500,000.

Step 1: Understanding Scientific Notation

The expression \(5 \times 10^{5}\) is given in scientific notation, where 5 is the coefficient and \(10^5\) describes the power of ten. Scientific notation is used to express large or small numbers in a concise way.

Step 2: Identify the Power of 10

In the expression \(5 \times 10^{5}\), the power of ten is 5. This means that we need to multiply 5 by 10 raised to the power of 5, which is 100,000.

Step 3: Perform the Multiplication

Calculate \(5 \times 10^{5}\) as follows: multiply 5 by 100,000. This involves shifting the decimal point of 5 five places to the right.

Step 4: Write the Standard Form

The result of the multiplication \(5 \times 100,000\) equals 500,000. Therefore, the number in standard form is 500,000.

Key Concepts

Standard Form

Standard form is a way of writing numbers that is both simple and precise. It is often used to express very large or very small numbers in a more manageable way.

• Standard form is also known as "Scientific Notation".
• It is useful for simplifying calculations and making data easier to read.

To write a number in standard form, it is expressed as a number between 1 and 10 multiplied by a power of ten. This means that you adjust the number to fall within this range, moving the decimal point as needed, and then count how many places you moved it to establish the power of ten. For example, the number 500,000 in standard form is written as \( 5 \times 10^{5} \), where 5 is the coefficient and \( 10^{5} \) is the power of ten.

Power of Ten

The power of ten is a crucial element in standard form. It determines how many times we multiply the coefficient by ten. Here are some important points to keep in mind:

• The power of ten is denoted by an exponent, which tells us the number of times to multiply the base (10) by itself.
• In \( 5 \times 10^{5} \), the power is 5, meaning 10 is multiplied by itself five times: 10 x 10 x 10 x 10 x 10.
• Each multiplication by ten shifts the decimal point to the right by one place.

Visualize it as a shortcut that streamlines the way we represent large numbers. So, rather than writing out zeros, the exponent handles the heavy lifting.

Multiplication

Multiplication is the next step after identifying the power of ten in scientific notation. It involves multiplying the coefficient (the number in front) by the expanded version of the power of ten. Let's breakdown the multiplication:

• First, understand that \( 10^{5} \) means 100,000. This is derived from multiplying ten by itself five times.
• Next, multiply the coefficient, in this case, 5, by 100,000. This is done by shifting the decimal point of 5 to the right five places.
• The result of \( 5 \times 100,000 \) is 500,000.

This process of multiplication not only gives us the number in standard form but also shows how powerful and efficient scientific notation can be in simplifying complex calculations.