Problem 87 - College Algebra

Question

Perform the indicated operation and express the answer in decimal notation. $$ \frac{6.3 \times 10^{3}}{3 \times 10^{5}} $$

Step-by-Step Solution

Answer

The result, in decimal notation, of the given operation is 0.021

Step 1: Separate into two divisions

Separate the given expression into two individual division operations, one for the decimal components and one for the exponential components. This can be written as \((6.3 / 3) \times (10^{3} / 10^{5}) \)

Step 2: Simplify the decimal division

Perform the division operation 6.3 divided by 3. This will equate to 2.1.

Step 3: Simplify the exponential division

For division operation, subtract the exponent in the denominator from the exponent in the numerator. So, calculate \(10^{3-5}\), which simplifies to \(10^{-2}\).

Step 4: Express in decimal notation

Finally, multiply the results from step 2 and step 3 together. In other words, multiply 2.1 by \(10^{-2}\). This will give 0.021, which is the solution in decimal notation.

Key Concepts

Exponential Division

When you deal with exponential division, you are working with numbers in powers of ten, often seen in scientific notation. Performing exponential division means subtracting the exponent of the denominator from the exponent of the numerator. For example, consider \( \frac{10^3}{10^5} \). Here, you subtract the exponent 5 from 3, giving you \( 10^{-2} \). This simplifies your calculations significantly, as it reduces the complexity of multiplying large numbers. Remember, the rule for dividing powers of the same base is to subtract exponents: \( \frac{a^m}{a^n} = a^{m-n} \). Using exponential division is key to managing and simplifying tasks in mathematics and science efficiently.

Decimal Division

Dividing decimal numbers can be straightforward. You simply perform the division as you would with whole numbers, ensuring that you place the decimal point correctly in the result. When dividing, for example, 6.3 by 3, start from the highest value digit and work through the number, placing the decimal point in the quotient directly above where it appears in the dividend. Here, 6 divided by 3 gives 2, then consider the decimal: 3 into 0.3 occurs 0.1 times, totaling a quotient of 2.1. Break the numbers into smaller parts if needed, and carefully calculate each segment. Maintaining precision with decimal points ensures accuracy in your results.

Scientific Notation

Scientific notation is a way to express very large or very small numbers efficiently. A number in scientific notation is written as the product of a number (greater than or equal to 1 and less than 10) and a power of ten. For example, 6300 can be written as \( 6.3 \times 10^3 \). The purpose of scientific notation is to simplify calculations and avoid writing long strings of zeros. When converting back to decimal form, adjust the decimal by the exponent's power: a negative exponent moves the decimal point left, and a positive exponent moves it right. Multiplying \( 2.1 \times 10^{-2} \) shifts the decimal point two places left, resulting in 0.021. Mastery of scientific notation is invaluable in fields like physics, astronomy, and engineering, where it facilitates the handling of extreme quantities succinctly.