Question

Divide without using a calculator. Give your answer in scientific notation. $$\left(8 \times 10^{4}\right) \div\left(2 \times 10^{2}\right)$$

Short answer

The result is \(4 \times 10^{2}\).

Step-by-step solution

Divide the coefficients

First, divide the numeric parts (also called coefficients) of the scientific notations without considering the powers of ten. So, divide 8 by 2.

Subtract the exponents

Next, subtract the exponent of the divisor from the exponent of the dividend. In this case, subtract the exponent 2 from the exponent 4.

Combine the results in scientific notation

Combine the quotient of the coefficients with the result of subtracting the exponents to express the final answer in scientific notation.

Key concepts

Dividing Coefficients

Understanding dividing coefficients is foundational when working with scientific notation in arithmetic problems. Coefficients in scientific notation represent the significant figures of a number, while the exponent signifies the order of magnitude. When dividing two numbers expressed in scientific notation, you start by dividing these coefficients as if they were ordinary numbers. For example, if you have the problem \(8 \times 10^{4}) \div (2 \times 10^{2})\), first focus solely on the numbers 8 and 2.

Dividing 8 by 2 is straightforward and yields a quotient of 4. Note that this operation does not involve the power of ten at this stage. Making sure you correctly divide the coefficients is crucial for achieving the correct magnitude in your final answer. Mistakes in this step could lead to vastly incorrect results due to the exponential nature of scientific notation.

Subtracting Exponents

With the coefficient division complete, we move on to subtracting exponents, which manages the scale represented by scientific notation. When you divide numbers in scientific notation, you subtract the exponent of the denominator from the exponent of the numerator. This step adheres to the laws of exponents, which dictate that when we divide powers with the same base, we subtract the exponents.

In the given problem, \(8 \times 10^{4}) \div (2 \times 10^{2})\), the exponent from the denominator (2) is subtracted from the exponent of the numerator (4), resulting in \(10^{4-2}\) or \(10^{2}\). It’s essential to understand that this operation simplifies the powers of ten that multiply the coefficients, effectively scaling down or scaling up the quotient according to the laws of indices.

Scientific Notation Arithmetic

Combining the previously discussed steps culminates in what's known as scientific notation arithmetic. It's an essential tool in mathematics for handling very large or very small numbers efficiently. After dividing the coefficients and subtracting the exponents separately, you write the result in the compact form of scientific notation. The final result represents the same number as if it were written out in full decimal form but in a much more manageable format.

The solution to our example, \(8 \times 10^{4}) \div (2 \times 10^{2})\), after performing the earlier steps, is \(4 \times 10^{2}\). The answer, now in scientific notation, has been obtained by combining the result of the coefficient division (4) with the result of subtracting the exponents (\(10^{2}\)). It is crucial to represent the final answer correctly in scientific notation for clarity and accuracy. Thus, mastering the complete process of scientific notation arithmetic simplifies calculations and helps in accurately representing and understanding numbers across diverse scientific fields.