Problem 55 - Technical Mathematics with Calculus

Question

Divide without using a calculator. Give your answer in scientific notation. $$\left(3 \times 10^{3}\right) \div\left(6 \times 10^{5}\right)$$

Step-by-Step Solution

Answer

\(5 \times 10^{-3}\)

Step 1: Simplify the Constants

Divide the constants 3 and 6, which are the coefficients of the powers of 10. The division of 3 by 6 gives 0.5.

Step 2: Apply the Laws of Exponents

Subtract the exponent of the divisor from the exponent of the dividend to simplify the expression \(10^{3} \/ 10^{5} = 10^{3-5} = 10^{-2}\).

Step 3: Combine the Results

Combine the simplified constants and the result of the exponent operation to obtain \(0.5 \times 10^{-2}\).

Step 4: Convert to Scientific Notation

Convert \(0.5\) to scientific notation by expressing it as \(5 \times 10^{-1}\). Thus, the final result is \(5 \times 10^{-3}\).

Key Concepts

Laws of Exponents

Understanding the laws of exponents is crucial when dealing with scientific notation, especially during division. The laws are simple rules that describe how to manipulate exponents on numbers. One fundamental law that applies to division is the rule which states that when you divide powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

For example, in the expression \(10^{3} \div 10^{5}\), you would subtract the exponent of the denominator (5) from the exponent of the numerator (3) to get \(10^{-2}\). This powerful law makes simplifying expressions with exponents much easier and is a key aspect of working effectively with scientific notation.

Another key exponent law is to remember that any number raised to the power of zero is 1 (except for the number zero itself), and raising a number to a negative exponent results in the reciprocal of that number raised to the positive of that exponent. For instance, \(10^{-2}\) would equal \(\frac{1}{10^{2}}\) or \(0.01\).

Simplifying Expressions

Simplifying expressions is the process of reducing an expression to its most basic form without changing its value. When dealing with scientific notation, simplification often involves managing coefficients (numbers in front of \(10\)) and adjusting exponents accordingly.

Consider the division exercise provided: when dividing the coefficients 3 by 6, you’re simplifying the numeric part of the scientific notation before dealing with the exponents. It is essential to perform these steps separately yet methodically to maintain the integrity of the expression.

After handling the coefficients, you simplify the exponential part using the laws of exponents as discussed earlier. The goal is to end up with a single coefficient multiplied by a power of ten, which is easy to understand and use. It's a lot like organizing a cluttered room: sort out each item before putting everything back in place.

Scientific Notation Conversion

Converting numbers to and from scientific notation is a vital skill in science and mathematics, allowing us to express very large or very small numbers compactly. The conventional form of scientific notation includes a coefficient between 1 and 10, followed by \(10\) raised to an exponent.

In step 4 of the solution, the coefficient of \(0.5\) is not between 1 and 10, so it must be converted to fit the convention of scientific notation. To do this, the number is multiplied by 10 and the exponent is decreased by 1. This moves the decimal one place to the right. Hence, \(0.5\) becomes \(5 \times 10^{-1}\).

By applying this process, you can readily convert any number into scientific notation, enabling you to handle and communicate large-scale calculations and measurements with ease. After the conversion, the expression can be easily used in further mathematical or scientific analysis.