Problem 67 - Elementary and Intermediate Algebra

Question

Use scientific notation to perform the calculations. Give all answers in scientific notation and standard notation. \(\frac{9.3 \times 10^{2}}{3.1 \times 10^{-2}}\)

Step-by-Step Solution

Answer

\(3.0 \times 10^4\) in scientific notation; 30000 in standard notation.

Step 1: Identify the Numbers in Scientific Notation

First, identify the numbers involved in the calculation. You have \(9.3 \times 10^{2}\) and \(3.1 \times 10^{-2}\).

Step 2: Divide the Coefficients

Divide the coefficients \(9.3\) by \(3.1\). When you perform the division, you get \(9.3 / 3.1 = 3.0\).

Step 3: Subtract the Exponents

For scientific notation division, subtract the exponent in the denominator from the exponent in the numerator: \(2 - (-2) = 2 + 2 = 4\).

Step 4: Form the Scientific Notation Result

Combine the result of Step 2 and Step 3. The result from Step 1 is \(3.0\), and from Step 3 is \(10^4\). Therefore, the scientific notation result is \(3.0 \times 10^4\).

Step 5: Convert to Standard Notation

To convert \(3.0 \times 10^4\) into standard notation, move the decimal point in \(3.0\) four places to the right, resulting in 30000.

Key Concepts

Division of Scientific Notation

When dividing numbers in scientific notation, the process involves two main steps: dealing with their coefficients and their exponents. Scientific notation is a way to express large or small numbers conveniently by multiplying a decimal number (coefficient) by a power of ten. For division:

• First, you handle the coefficients. Look at the two numbers: \(9.3\) and \(3.1\). Divide these as you would any regular numbers, capturing the essence of scientific computation.
• Then, tackle the exponents. The rule for division is to subtract the exponent in the denominator from the exponent in the numerator. In our example, this means calculating \(2 - (-2)\), which simplifies to \(+2 + 2 = 4\).

This ensures that you maintain the integrity of the original numbers while simplifying the operation.

Exponents in Scientific Notation

Exponents are the pivotal part of scientific notation. They show just how large or small a number truly is by referring to the power of ten by which the coefficient is multiplied. Here’s how they work with division:

• When you divide numbers with exponents, you're effectively reducing a number to its simplest form by utilizing the rules of exponents. For example, if you see terms such as \(10^a\) divided by \(10^b\), the division results in \(10^{a-b}\).
• In the given example, the numerator has an exponent of \(2\) and the denominator has \(-2\). By subtracting \(-2\) from \(2\), it transforms the operation to addition: \(2 - (-2) = 4\).

These exponent manipulations help condense potentially large calculations into simpler evaluations.

Conversion to Standard Notation

Converting scientific notation back to standard notation is a straightforward process. It turns the compact form into its full numerical representation, useful for clear understanding:

• Let's consider the result \((3.0 \times 10^4)\). The exponent \(4\) tells you to move the decimal 4 places to the right.
• For each step moving right, essentially it's like multiplying by 10, turning \((3.0)\) into 3000, then maintaining movement until 30000.

This shifts the compact scientific notation into a number that's instantly recognizable in its full magnitude.