Problem 54 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \left(3 \times 10^{1}\right)\left(3 \times 10^{4}\right) $$

Step-by-Step Solution

Answer

The result is \(9 \times 10^5\) in scientific notation and 900,000 in standard form.

Step 1: Separate the Elements

The expression \( \left(3 \times 10^{1}\right)\left(3 \times 10^{4}\right) \) consists of two parts, namely \(3 \times 10^1\) and \(3 \times 10^4\). These need to be multiplied individually by their respective coefficients (3 and 3) and their powers of 10 (\(10^1\) and \(10^4\)).

Step 2: Multiply the Coefficients

First, multiply the coefficients: \(3 \times 3 = 9\).

Step 3: Add the Exponents

Next, handle the powers of 10. When multiplying like bases, you add the exponents: \(10^1 \times 10^4 = 10^{1+4} = 10^5\).

Step 4: Combine Results

Combine the results from previous steps: \( 9 \times 10^5 \).

Step 5: Convert to Standard Form

Convert \(9 \times 10^5\) to standard form by moving the decimal point five places to the right, which gives \(900,000\).

Step 6: Confirm Scientific Notation

Ensure the scientific notation \(9 \times 10^5\) is correct, where the coefficient is between 1 and 10.

Key Concepts

Multiplication of Exponents

When you multiply terms with exponents that have the same base, you should add the exponents together. This is a fundamental rule in handling exponents. In the given expression, both terms contain a base of 10: \(10^1\) and \(10^4\). To multiply them:

• Keep the base 10.
• Add the exponents: \(1 + 4 = 5\).

This gives us \(10^5\).
Always remember this rule: multiplying with the same base is about adding the powers. It's helpful because it simplifies expressions dramatically, especially when dealing with larger calculations or multiple steps.

Standard Form Conversion

To convert a number from scientific notation to standard form, you adjust the placement of the decimal based on the exponent. Here, our expression in scientific notation is \(9 \times 10^5\).

• The exponent 5 tells us how many places to shift the decimal to the right.
• Starting from 9.0, moving five places right gives us 900,000.

Always remember that a positive exponent shifts the decimal to the right, turning smaller decimal figures into larger whole numbers. When numbers are expressed in scientific notation, this method helps quickly translate them into standard numerical values that are easy to read and use.

Multiplication of Numbers in Scientific Notation

Handling multiplication in scientific notation may look complex, but it's straightforward when you break it down. In the initial expression \((3 \times 10^1)(3 \times 10^4)\) you:

• Start by multiplying the numerical coefficients, here 3 \(\times\) 3 equals 9.
• Next, multiply the powers of ten, where we add the exponents, resulting in \(10^5\).
• Combine both results to get \(9 \times 10^5\).

This process showcases the power of simplifying complex problems using scientific notation by separating and dealing with the individual components. It's an efficient method that saves time and makes calculations manageable, even for very large or very small numbers. Remember, breaking the problem into smaller parts is key to clarity.