Problem 48 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation without using a calculator. Write the result in scientific notation. $$ \left(9 \times 10^{-6}\right)\left(2 \times 10^{4}\right) $$

Step-by-Step Solution

Answer

The answer to this multiplication in scientific notation is \(1.8 \times 10^{-1}\)

Step 1: Multiply coefficients

Multiply the coefficients of the numbers in scientific notation, these are the numbers before the exponential term, which in this case are 9 and 2. So the multiplication of these two numbers is \(9 \times 2 = 18\).

Step 2: Add the exponents

Add the exponents of the same base (which is 10 in this case). Our exponents are -6 and 4, so when added, the result is \(-6 + 4 = -2\).

Step 3: Write the Result in Scientific Notation

The result, in scientific notation, after multiplication of coefficients and addition of exponents, becomes \(18 \times 10^{-2}\).

Step 4: Adjust to Correct Form of Scientific Notation

The result \(18 \times 10^{-2}\) is not in correct scientific notation format because the first part of the number (i.e., coefficient, which is 18 in this case) is not between 1 and 10. So we move the decimal point of 18 one place to the left to make it 1.8 and increase the exponent by 1 (which is -2) to make it -1. So the final answer in proper scientific notation is \(1.8 \times 10^{-1}\).

Key Concepts

Multiplying Coefficients in Scientific Notation

When multiplying numbers in scientific notation, start by focusing on the coefficients. These are generally the numeric parts before the exponent. For instance, if you have \(9 \times 10^{-6}\) and \(2 \times 10^{4}\), the coefficients here are 9 and 2 respectively.
To multiply the coefficients, simply multiply these numbers as you would with any standard numbers: \(9 \times 2 = 18\). It's straightforward arithmetic, but make sure to maintain precision, especially with larger or decimal coefficients.

Remember:

• The product of the coefficients will be part of the final result in scientific notation.
• If the result is not between 1 and 10, further adjustment is needed to fit scientific notation standards.

Adding Exponents

Once the coefficients are multiplied, the next step is to add the exponents. This process only works when the base of the exponents is the same, as it is the case with scientific notation where the base is always 10.
For example, \(9 \times 10^{-6}\) \times \(2 \times 10^{4}\) means you'll be adding the exponents -6 and 4 together: \( -6 + 4 = -2\).

Key Points:

• Make sure the exponents have the same base before adding them; in scientific notation, the base is 10.
• The addition of exponents reflects the multiplication of the quantities represented by the scientific notation.

Adding exponents streamlines the multiplication process into one elegant operation for numbers in scientific notation.

Scientific Notation Rules

Understanding the rules of scientific notation is crucial for working with very large or very small numbers efficiently.
The standard form of a number in scientific notation requires:

• A coefficient between 1 and 10 (including 1, but not 10).
• An exponent that is an integer.
• The base of 10.

When multiplying two numbers in scientific notation, these rules mean that after the initial operations, adjustments may be required to ensure the result fits the form \(a \times 10^{n}\), where \(1 \leq a < 10\) and \(n\) is an integer. Following these rules keeps numbers manageable and makes calculations with them more systematic and less prone to error.

Adjusting Scientific Notation Format

If the multiplication of coefficients results in a number that is not between 1 and 10, you'll need to adjust it to fit proper scientific notation.
Take our product from earlier, \(18 \times 10^{-2}\). Here, the coefficient 18 is not between 1 and 10. To adjust, we can shift the decimal point one position to the left, turning it into 1.8. Consequently, we must increase the exponent by 1 to balance the equation, resulting in \(1.8 \times 10^{-1}\) being the correct scientific notation.

Keep in Mind:

• Each shift of the decimal point to the left increases the exponent by 1.
• Conversely, a shift to the right decreases the exponent by 1.
• Adjustment ensures the coefficient remains between 1 and 10, preserving the integrity of scientific notation.

Following these steps correctly results in easily comprehensible and universally recognizable scientific notation.