Problem 63 - Technical Mathematics with Calculus

Question

Perform the following computations. Display your answer in scientific notation. $$\left(5.6 \times 10^{2}\right)\left(3.1 \times 10^{-1}\right)$$

Step-by-Step Solution

Answer

The product is in scientific notation: \(1.736 \times 10^{2}\).

Step 1: Multiplying the coefficients

Multiply the numerical coefficients: 5.6 and 3.1. The product is: 5.6 * 3.1.

Step 2: Adding the exponents

Since the bases (10) of the exponents are the same, you can add the exponents. Add the exponents 2 and -1 to get the new exponent.

Step 3: Calculate the product

Compute the multiplication from Step 1 and the addition from Step 2 to obtain the result in scientific notation.

Step 4: Adjusting the scientific notation

If necessary, adjust the result so that there is only one non-zero digit to the left of the decimal point which is the standard form of scientific notation.

Key Concepts

Multiplying Coefficients in Scientific Notation

In scientific notation, numbers are typically represented in the form of a product of a coefficient and a power of ten. This concise format is valuable when dealing with very large or very small numbers. The key to multiplying numbers in scientific notation is to handle the coefficients and the exponents separately.

When multiplying coefficients, you simply multiply the numbers as though you are calculating a regular multiplication problem. It is essential to keep in mind that the coefficient should be a number between 1 and 10. For instance, if we have two numbers in scientific notation, such as \(5.6 \times 10^{2}\) and \(3.1 \times 10^{-1}\), we start by multiplying the coefficients 5.6 and 3.1 which gives us 17.36. This multiplication is the first step in combining the two numbers in scientific notation.

Adding Exponents

After multiplying the coefficients, the next step involves the powers of ten. Here, we utilize one of the fundamental rules of exponents: when multiplying powers with the same base, you add the exponents.

Using our example, we have exponents 2 and -1 from the numbers \(5.6 \times 10^{2}\) and \(3.1 \times 10^{-1}\), respectively. To combine them, we add 2 to -1, resulting in an exponent of 1. This leaves us with \(17.36 \times 10^{1}\). It’s crucial that students remember this rule, as it is a key component of multiplying numbers in scientific notation and is widely applicable across various disciplines in mathematics and science.

Scientific Notation Standard Form

Upon completion of the multiplication and addition of exponents, the final step is to express the number in the standard scientific notation form. The standard form requires that the coefficient be between 1 and 10, with only one non-zero digit before the decimal point.

In our exercise, the multiplication gave us 17.36, which is not between 1 and 10. To convert it to standard form, we adjust the coefficient by moving the decimal point one digit to the left, which also changes the exponent. The coefficient becomes 1.736, and since we moved the decimal one place, we have to increase the exponent by 1. Thus, the final answer in standard scientific notation is \(1.736 \times 10^{2}\). The adjustment ensures that the result is correctly formatted and easy to read, adhering to the conventions of scientific notation.