Question

Write each of the following numbers in standard scientific notation. a. \(1 / 0.00032\) b. \(10^{3} / 10^{-3}\) c. \(10^{3} / 10^{3}\) d. \(1 / 55,000\)

Short answer

a. \(1 / 0.00032 = 3.125 \times 10^{3}\) b. \(10^{3} / 10^{-3} = 10^{6}\) c. \(10^{3} / 10^{3} = 10^{0}\) d. \(\frac{1}{55,000} = 1.81818... \times 10^{-5}\)

Step-by-step solution

Answer

\(1 / 0.00032 = 3.125 \times 10^{3}\) #b. \(10^{3} / 10^{-3}\)# #Step 1: Simplify the expression# To find the result, divide \(10^{3}\) by \(10^{-3}\): \(10^{3} \div 10^{-3} = 10^{3 + 3}\) #Step 2: Convert to scientific notation# Since the expression is already in scientific notation, no conversion is necessary: \(10^{6}\)

Answer

\(10^{3} / 10^{-3} = 10^{6}\) #c. \(10^{3} / 10^{3}\)# #Step 1: Simplify the expression# To find the result, divide \(10^{3}\) by \(10^{3}\): \(10^{3} \div 10^{3} = 10^{3 - 3}\) #Step 2: Convert to scientific notation# Since the expression is already in scientific notation (and equal to 1), no conversion is necessary: \(10^{0}\)

Answer

\(10^{3} / 10^{3} = 10^{0}\) #d. 1 / 55,000# #Step 1: Simplify the expression# To find the result, divide 1 by 55,000: \(1 \div 55,000 = 0.0000181818...\) #Step 2: Convert to scientific notation# To convert 0.0000181818... to standard scientific notation, we need to shift the decimal point 5 places to the right: \(1.81818... \times 10^{-5}\)

Answer

\(\frac{1}{55,000} = 1.81818... \times 10^{-5}\)

Key concepts

Exponential Notation

Exponential notation, also known as scientific notation, is a concise way of representing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used in science and engineering to simplify calculations and clearly display figures.

An exponential notation consists of two parts: a coefficient and a base raised to an exponent. The coefficient is a number greater than or equal to 1 and less than 10, and the base is usually 10. The exponent indicates the number of places the decimal point has moved.

For example, the number 2,300 can be written in exponential notation as \( 2.3 \times 10^3 \). Here, the coefficient is 2.3 and the exponent is 3, indicating that the decimal has shifted 3 places to the right. This notation is particularly useful for very large or very small numbers, such as the speed of light (approximately \( 3 \times 10^8 \) meters per second) or the charge of an electron (approximately \( -1.6 \times 10^{-19} \) coulombs).

Mathematical Operations

Performing mathematical operations with numbers in exponential notation follows specific rules that can simplify the calculation process.

Multiplying numbers requires you to multiply the coefficients and add the exponents if the base is the same. For example, \( (2 \times 10^3) \times (3 \times 10^2) = 6 \times 10^{3+2} = 6 \times 10^5 \).

When dividing numbers, you divide the coefficients and subtract the exponents, again assuming the base is the same. As illustrated in the exercise solution b, \( 10^3 / 10^{-3} = 10^{3 - (-3)} = 10^{3+3} = 10^6 \), showcasing how to perform the operation within exponential notation.

These operations facilitate working with very large or small numbers because they prevent unwieldy string of zeros in calculations, which can be confusing and prone to error. Knowing how to manipulate exponents becomes crucial when dealing with scientific data.

Scientific Notation Conversion

Converting a number to scientific notation often requires moving the decimal point to create a new number from 1 up to 10, and then indicating how many places the decimal point was moved using an exponent on the base 10.

When converting a large number to scientific notation, the decimal point moves to the left, and the exponent on 10 is positive. Contrastingly, for small numbers, as demonstrated in exercise solution d, where the starting number is less than one (\( 1/55,000 \)), the decimal point moves to the right, resulting in a negative exponent (\( 1.81818... \times 10^{-5} \)).

Effective conversion to scientific notation allows for a standardized format which is universally recognized and facilitates an easier way to compare numbers, estimate sizes or scales, and perform mathematical operations. It's an essential skill for students in many fields of study, particularly those involving quantitative analysis or data that spans a wide range of magnitudes.