Problem 55 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \left(5 \times 10^{2}\right)\left(7 \times 10^{-4}\right) $$

Step-by-Step Solution

Answer

The expression evaluates to \(3.5 \times 10^{-1}\) in scientific notation and 0.35 in standard form.

Step 1: Multiply the coefficients

First, multiply the numerical coefficients in the expression: 5 and 7. Thus, we calculate: \[ 5 \times 7 = 35 \]

Step 2: Add the exponents in the powers of ten

Next, we need to work with the powers of ten. According to the laws of exponents, when multiplying numbers with the same base, we add their exponents.Hence, we combine the exponents: \[ 10^{2} \times 10^{-4} = 10^{(2 + (-4))} = 10^{-2} \]

Step 3: Combine results

After multiplying the coefficients and adding the exponents, we combine the results from Steps 1 and 2.The expression simplifies to: \[ 35 \times 10^{-2} \] This is the result in scientific notation.

Step 4: Convert to standard form

Finally, convert the scientific notation into standard form. Moving the decimal 2 places to the left (since the exponent is -2), we have: \[ 0.35 \]

Key Concepts

Understanding Exponents

Exponents are a shorthand way of indicating that a number should be multiplied by itself a specific number of times. They are expressed as a small number placed as a superscript to the right of a base number. In the expression \(10^2\), the number 10 is the base, and the 2 is the exponent. This means you multiply 10 by itself once, as follows:

• \(10^2 = 10 \times 10 = 100\)

The rules of exponents are crucial when performing operations like multiplication or division with them. When multiplying numbers with the same base, you add the exponents. For example:

• \(10^a \times 10^b = 10^{a+b}\)

This rule makes calculations simpler and faster, especially when working with large numbers. Understanding these rules also helps in converting between scientific notation and standard form.

Converting to Standard Form

Standard form is the way we typically write numbers, without any multiplication component. It's a straightforward decimal representation of the number. To convert from scientific notation, one needs to adjust the decimal point based on the exponent.

• A positive exponent moves the decimal to the right.
• A negative exponent moves the decimal to the left.

Let's consider our earlier result in scientific notation: \(35 \times 10^{-2}\). The \-2 exponent indicates moving the decimal two places to the left, resulting in:

• \(0.35\) in standard form.

Converting between scientific notation and standard form is essential because it allows numbers to be expressed more naturally, especially useful when the number should be presented in common, everyday usage.

Multiplying Decimals Effectively

Multiplying decimals involves a few simple steps. First, temporarily ignore the decimal places and multiply the numbers as if they were whole numbers. Then at the end, adjust the result by placing the decimal point appropriately. Consider multiplying \(0.5\) and \(0.7\). First, multiply 5 and 7:

• \(5 \times 7 = 35\)

Next, count the total number of decimal places in the original numbers (2 places), and adjust the decimal point in the answer:

• The result becomes \(0.35\).

In cases involving scientific notation, this method aids significantly, especially when simplifying tasks into smaller parts like focusing first on multiplying coefficients separate from the powers of ten. It transforms what might be a complex problem into a simple and manageable one.