Problem 68 - Elementary and Intermediate Algebra

Question

Use scientific notation to perform the calculations. Give all answers in scientific notation and standard notation. \(\frac{7.2 \times 10^{6}}{1.2 \times 10^{8}}\)

Step-by-Step Solution

Answer

The answer is \(6 \times 10^{-2}\) in scientific notation and 0.06 in standard notation.

Step 1: Divide the Coefficients

First, divide the numerical coefficients in the scientific notation expressions. Divide 7.2 by 1.2:\[\frac{7.2}{1.2} = 6\]

Step 2: Subtract the Exponents

Next, subtract the exponents of the powers of ten. Subtract 8 from 6:\[6 - 8 = -2\]

Step 3: Combine the Results

Multiply the result from dividing the coefficients by the power of ten with the adjusted exponent from the previous step:\[6 \times 10^{-2}\]

Step 4: Convert to Standard Notation

Convert the scientific notation result to standard notation by moving the decimal place of the coefficient according to the exponent:\[6 \times 10^{-2} = 0.06\]

Key Concepts

Division of Exponents

When you're performing calculations with numbers in scientific notation, one of the key steps is managing the exponents. The rule for division of exponents hinges on subtracting the exponents of numbers that share the same base. In the context of scientific notation, the base is always 10. For example, given an expression like \(\frac{10^6}{10^8}\), you simplify it by subtracting the exponents: \(6 - 8 = -2\). Therefore, \(\frac{10^6}{10^8} = 10^{-2}\). This rule makes it easier to deal with very large or very small numbers without writing out all the zeroes.

Standard Notation

Standard notation refers to writing numbers in the regular decimal form that we commonly use. In contrast to scientific notation, which is useful for expressing very large or very small numbers, standard notation presents numbers as they regularly appear. Converting a number from scientific to standard notation involves moving the decimal point in accordance with the exponent.
For example, if you have the number \(6 \times 10^{-2}\), you would convert it to 0.06 by moving the decimal point two places to the left because the exponent is -2. This conversion helps when you need to understand the actual scale or magnitude of the number in everyday terms.
When the exponent is positive, the decimal point moves to the right.

Coefficients in Scientific Notation

The coefficient in scientific notation is the number that is multiplied by a power of ten. It’s usually a decimal number greater than or equal to 1 and less than 10. To perform operations with scientific notation, you often start by handling the coefficients separately before addressing the powers of ten.
In the problem \(\frac{7.2 \times 10^6}{1.2 \times 10^8}\), the first step is to divide the coefficients, which are 7.2 and 1.2. This results in \(\frac{7.2}{1.2} = 6\).
The next step involves adjusting the exponents, as previously discussed. Focusing on the coefficients first simplifies handling the large numbers scientific notation usually involves.