Problem 100 - Elementary Algebra

Question

\(P=\left(3.0 \times 10^{-2} \mathrm{~kg}\right)\left(5.4 \times 10^{1} \frac{\mathrm{m}}{\mathrm{s}}\right)\)

Step-by-Step Solution

Answer

The product is \(16.2 \times 10^{-1}\) or \(1.62\).

Step 1: Understand the Problem

The problem requires finding the product of two numbers written in scientific notation. Specifically, we need to multiply a mass by a velocity.

Step 2: Write the Numbers in Scientific Notation

Identify the numbers given in the problem. The mass is given as \(3.0 \times 10^{-2} \text{ kg}\), and the velocity is given as \(5.4 \times 10^{1} \text{ m/s}\).

Step 3: Multiply the Coefficients

Multiply the coefficients (the numbers in front of the power-of-10 terms). \(3.0 \times 5.4 = 16.2\)

Step 4: Add the Exponents

According to the rules of exponents, when multiplying terms with the same base, add the exponents. We have: \(10^{-2} \times 10^{1} = 10^{-2 + 1} = 10^{-1}\)

Step 5: Combine the Results

Combine the product of the coefficients with the result of the exponent addition. \(16.2 \times 10^{-1}\)

Step 6: Rewrite the Scientific Notation

Ensure the final answer is in correct scientific notation. \(16.2 \times 10^{-1}\) can be written as \(1.62 \times 10^{0}\), but no further normalization is required here.

Key Concepts

scientific notation

Scientific notation is a way to express very large or very small numbers in a compact form. This makes it easier to read, write, and work with these numbers, especially in scientific contexts. It follows the format:

• A coefficient: a number, generally between 1 and 10.
• An exponent: indicates how many times the coefficient is multiplied by 10.

For example, in the number \(3.0 \times 10^{-2}\), 3.0 is the coefficient, and -2 is the exponent of 10. It means \(3.0 \times 0.01 = 0.03\). Understanding scientific notation is essential, as it allows you to easily perform mathematical operations like multiplication and division.

multiplication of scientific notation

When multiplying numbers in scientific notation, follow these two main steps:

• Multiply the coefficients: This means multiplying the numbers in front of the power of 10 terms.
• Add the exponents: This involves adding the powers of 10 together.

For instance, in the exercise, we have to calculate \(P=(3.0 \times 10^{-2} \mathrm{~kg})(5.4 \times 10^{1}\mathrm{m/s})\). First, multiply the coefficients: \(3.0 \times 5.4 = 16.2\). Secondly, handle the exponents by adding them: \(10^{-2} \times 10^{1} = 10^{-2+1} = 10^{-1}\). Finally, combine both results: \(16.2 \times 10^{-1}\).

exponent addition

Exponent addition is a crucial step in the multiplication of numbers in scientific notation. When two numbers with a common base (like 10) are multiplied, their exponents are added together. This follows the rule:
\[ a^m \times a^n = a^{m+n} \]In the example provided within the exercise, the exponents are -2 and 1. These are added to yield:
\[ 10^{-2} \times 10^{1} = 10^{-2+1} = 10^{-1} \]It is important to not just mechanically add the exponents but also understand why this rule works. Multiplying powers of 10 simplifies by shifting the decimal place according to the exponent. This streamlined process is what makes scientific notation powerful and efficient for handling large-scale multiplications or divisions.