Problem 97 - Algebra A Combined Function

Question

Evaluate each expression using exponential rules. Write each result in standard form. $$ \left(1.2 \times 10^{-3}\right)\left(3 \times 10^{-2}\right) $$

Step-by-Step Solution

Answer

The expression evaluates to 0.000036 in standard form.

Step 1: Break Down the Expression

Begin by writing out the expression \((1.2 \times 10^{-3})(3 \times 10^{-2})\) as a multiplication of coefficients and powers of ten. This gives us \[(1.2 \times 3) \times (10^{-3} \times 10^{-2})\].

Step 2: Multiply the Coefficients

Now multiply the coefficients:\[1.2 \times 3 = 3.6\].

Step 3: Add the Exponents

Next, use the product of powers rule for the exponents: when you multiply powers with the same base, you add the exponents. So, \[-3 + (-2) = -5\].

Step 4: Combine Results

Combine the results from steps 2 and 3 to write the expression in scientific form:\[3.6 \times 10^{-5}\].

Step 5: Convert to Standard Form

Standard form requires all the powers of ten be evaluated. Calculate:\[3.6 \times 10^{-5} = 0.000036\]. This is the standard form of the number.

Key Concepts

Multiplying Coefficients

When dealing with expressions that involve multiplication in scientific notation, the first step is to focus on the coefficients, which are the numbers in front of the powers of ten. In our example, we have the coefficients 1.2 and 3 from the expression

• \(1.2 \times 10^{-3}\)
• \(3 \times 10^{-2}\)

To multiply these coefficients, simply perform the multiplication as you would with any regular numbers:

• \(1.2 \times 3 = 3.6\)

This process simplifies the coefficients into a single number that will be used in combination with the powers of ten. Multiplying coefficients is crucial because it sets up the next steps in evaluating and rewriting the expression, ensuring that each component is combined correctly.

Adding Exponents

When you multiply numbers in scientific notation, it's not just the coefficients that need to be multiplied. You also need to deal with the exponents of the powers of ten. This involves a rule from exponents known as the "product of powers" rule. When numbers with the same base (

• in this case, the base is 10
• from powers of ten

) are multiplied together, you add their exponents. Our example has powers

• \(10^{-3}\)
• \(10^{-2}\)

Using the rule, you simply add these exponents:

• \(-3 + (-2) = -5\)

Adding exponents correctly is important as it determines the new power of ten in scientific notation. This step integrates seamlessly with the multiplied coefficients to produce a complete scientific notation ready to be simplified or further evaluated.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact format. It is represented by a number between 1 and 10 multiplied by a power of ten. In our example, after multiplying coefficients and adding exponents, we have:

• \(3.6 \times 10^{-5}\)

This is a scientific notation, where 3.6 is the coefficient, and \(10^{-5}\) is the power of ten part. Using scientific notation allows for easier calculation, comparison, and understanding of extremely large or small numbers. It is standardized, meaning everyone communicates these numbers consistently, avoiding any potential errors in interpretation due to their size. Recognizing numbers in scientific notation is essential for clarity in mathematics, science, and engineering fields.

Standard Form

After representing a number in scientific notation, you may need to convert it into standard form, which is the regular decimal format most of us are familiar with. To do this, evaluate the power of ten:

• multiply the coefficient by 10 raised to the power given

For our example, take \(3.6 \times 10^{-5}\) and shift the decimal point in the coefficient 3.6 five places to the left because the exponent is negative.This results in:

• \(0.000036\)

Standard form is practical for everyday usage because it presents numbers in a straightforward decimal format. It is particularly useful for visualizing the real-world scale of values, turning the abstractness of numbers into something more concrete and relatable. Being fluent in moving between scientific notation and standard form is a useful skill in both everyday mathematics and complex scientific computations.