Problem 53 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \left(4 \times 10^{3}\right)\left(2 \times 10^{5}\right) $$

Step-by-Step Solution

Answer

The result is \(8 \times 10^{8}\) in scientific notation and 800,000,000 in standard form.

Step 1: Multiply the Coefficients

Start by multiplying the numeric parts (coefficients) of the numbers together. Here, you need to multiply 4 by 2, which is fairly simple.\[4 \times 2 = 8\]

Step 2: Add the Exponents

Next, focus on the exponents of the base 10. When multiplying with the same base, you add the exponents together. This means adding the exponent 3 to the exponent 5.\[10^{3 + 5} = 10^{8}\]

Step 3: Combine the Coefficients and Exponents in Scientific Notation

Combine the result from Step 1 with the result from Step 2 to express the product in scientific notation. \[8 \times 10^{8}\]

Step 4: Convert to Standard Form

To convert the number from scientific notation to standard form, write it as a whole number. Here, "\(8 \times 10^{8}\)" translates to writing an 8 followed by eight zeros.\[800,000,000\]

Key Concepts

Exponents

Exponents are crucial in dealing with numbers that are multiplied together several times. They provide a shorthand way to express large repetitive multiplications. In the problem at hand, the exponent tells us how many times the base, 10, is multiplied by itself. For example, in the expression \(10^3\), the number 10 is multiplied by itself three times, i.e., \(10 \times 10 \times 10\).
This concept is slightly different when adding exponents, as seen in step 2 of the exercise. When you multiply numbers with the same base, you don't multiply the exponents; instead, you add them together. This is why \(10^3 \times 10^5\) becomes \(10^{3+5} = 10^8\). Logical tricks like these can simplify the process and make calculations much less daunting.

Multiplying Coefficients

When handling expressions in scientific notation, multiplying coefficients is one of the first steps. Coefficients are the numbers in front of the ten with an exponent, like 4 in \(4 \times 10^3\). To multiply coefficients, you simply multiply them as you would any other numbers. In our example, 4 and 2 are multiplied to get 8:

• 4 (from \(4 \times 10^3\))
• 2 (from \(2 \times 10^5\))

So, \(4 \times 2 = 8\).
This step focuses solely on the numbers without considering the powers of 10, making it a simple arithmetic step in the process of solving scientific notation problems.

Standard Form

Standard form is another way of saying a "normal" representation of a number, without exponents. This is used when converting from scientific notation, which is more compact, for ease in understanding or further calculations.
To convert from scientific notation to standard form, you multiply the coefficient by the power of ten, effectively moving the decimal point to the right. In this case, \(8 \times 10^8\), you write the number 8 followed by eight zeros:

• Equivalent to multiplying 8 by 100,000,000
• Resulting in 800,000,000

This form is often seen in fields like science and engineering where understanding exact values rather than compact notation is essential.

Algebra

Algebra involves working with symbols and numbers to express and solve equations. In the context of our problem, we are solving an expression using algebraic rules for multiplication and powers. The key is to keep organized:

• First, focus on multiplying coefficients to reduce complexity
• Then, handle the exponents by applying algebraic rules such as adding exponents when bases are the same

It is a process of handling one part of the operation at a time, ensuring simplicity, and reducing the chance for errors. Algebra serves as a powerful tool by providing a structured approach to dealing with various types of numerical operations and expressions.