Question

Divide without using a calculator. Give your answer in scientific notation. $$\left(8 \times 10^{-4}\right) \div 400,000$$

Short answer

\(2 \times 10^{-9}\)

Step-by-step solution

Convert the Divisor to Scientific Notation

First, convert the divisor (400,000) to scientific notation. Since 400,000 is the same as 4 followed by 5 zeros, it can be written as 4 x 10^5 in scientific notation.

Write the Division in Terms of Scientific Notation

Rewrite the division, with both the dividend (8 x 10^-4) and the divisor now in scientific notation (4 x 10^5), as a fraction: \[(8 \times 10^{-4}) \div (4 \times 10^{5})\].

Divide the Coefficients and Subtract the Exponents

Divide the coefficients (8 divided by 4) and subtract the exponent in the divisor from the exponent in the dividend (-4 - 5), remembering that subtracting a positive is the same as adding a negative. This gives a new coefficient of 2 and a combined exponent of (-4 - (5)).

Simplify the Exponent

Combine the exponents. Since we are dividing powers with the same base, we subtract the exponents: (-4 - 5) = -9.

Write the Final Answer in Scientific Notation

The final answer in scientific notation is thus \(2 \times 10^{-9}\).

Key concepts

Mathematical Computations Without a Calculator

Performing mathematical computations without the aid of a calculator may seem daunting, especially in an era where technology is ever-present. However, there are several advantages to mastering this skill. It not only reinforces the understanding of fundamental arithmetic principles but also enhances problem-solving abilities and provides the satisfaction of solving problems manually.

For division without a calculator, start by breaking down complex numbers into simpler forms and looking for patterns and shortcuts. For instance, when dividing large numbers, consider converting them into scientific notation, which simplifies large or very small numbers into a product of a number between 1 and 10 and a power of ten. Additionally, when dividing with whole numbers, look for divisible patterns, such as if a number ends in 0, it is divisible by 10. This can make mental division more feasible.

Scientific Notation Arithmetic

Scientific notation is a method used to express very large or very small numbers more succinctly. It is composed of two parts: a coefficient and an exponent. The coefficient is a decimal number between 1 and 10, and the exponent is a power of 10. Arithmetic with scientific notation follows the standard rules of arithmetic but requires special attention to the exponents when multiplying or dividing.

For division, as showcased in our original exercise, each part of the scientific notation is dealt with separately. The coefficients are divided just like any regular numbers, and the exponents are manipulated according to the rules of exponents—specifically, subtracting in the case of division. A helpful tip is to always ensure both numbers are in scientific notation before performing arithmetic operations. This consistency allows for a clearer understanding and easier computation.

Subtracting Exponents

When it comes to exponents, subtracting them is a crucial concept in mathematical computations, particularly in scientific notation arithmetic. The rule is straightforward: when you divide two powers with the same base, you subtract the exponents. This process is illustrated when dividing numbers in scientific notation.

To subtract exponents, simply take the exponent from the dividend (the number being divided) and subtract the exponent of the divisor (the number by which you are dividing). Keep in mind that when subtracting a positive number, it is equivalent to adding its negative.

For example, in the scientific notation \(8 \times 10^{-4}) \/ (4 \times 10^{5})\), you subtract the exponent 5 from the exponent -4, yielding an exponent of -9 after simplification. Remember, this rule only applies when the base of the exponents is the same. This principle is especially useful for scientific and engineering calculations, where dealing with numbers of different magnitudes is common.