Problem 3 - Technical Mathematics with Calculus

Question

Write each power of 10 as a decimal number. $$10^{-5}$$

Step-by-Step Solution

Answer

The decimal representation of \(10^{-5}\) is 0.00001.

Step 1: Understanding Negative Exponents

A negative exponent indicates that the base should be taken as a reciprocal and then raised to the opposite positive power. For the base 10, this means that we must divide 1 by 10 raised to the opposite positive power. In this case, we will find the value of 10 raised to the power of 5, and then take the reciprocal.

Step 2: Calculating Positive Exponent

First, calculate 10 raised to the power of 5: \(10^5 = 100000\). This is because 10 multiplied by itself 5 times equals 100000.

Step 3: Finding the Reciprocal

Since we need the reciprocal of \(10^5\), we flip the fraction to get the decimal representation of \(10^{-5}\). We know that \(10^{-5} = 1/(10^5)\). In decimal form, we'll need to place the decimal point in such a way that we add four zeros before placing the 1, which gives us 0.00001.

Key Concepts

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact form. It consists of two parts: a coefficient and a power of 10. The coefficient is a number greater than or equal to 1 but less than 10, and it is multiplied by 10 raised to an exponent. For example, the scientific notation for 0.00001 is written as \(5 \times 10^{-5}\).

This notation is particularly useful in fields like science and engineering, where such numbers occur frequently. Understanding scientific notation aids in working with powers of 10 and negative exponents, interpreting data, and converting between forms of representation. It's essential to recognize that the exponent provides information about the size of the number—negative exponents for numbers less than one, positive exponents for numbers greater than one.

Decimal Representation

Decimal representation refers to writing numbers in base 10, using digits 0 through 9 and a decimal point to indicate fractions. For example, the number \(10^{-5}\), when written in decimal form, becomes 0.00001. This is a very straightforward way of representing numbers and is the most familiar form for everyday use.

To convert a negative power of 10 to decimal, you move the decimal point to the left for each negative exponent increase. Therefore, the further left the decimal is shifted, the smaller the number becomes. This concept links directly to understanding negative exponents, and grasping this connection is invaluable for school mathematics and beyond.

Reciprocal of a Power

The reciprocal of a power, specifically when talking about a base of 10, is the inverse of that power of 10. For a number \(10^n\), its reciprocal is \(10^{-n}\), which means 1 divided by \(10^n\).

For example, the reciprocal of \(10^5\) is \(10^{-5}\) or \(1/(10^5)\). This can be visualized as taking the original value—where the base is multiplied by itself a certain number of times—and instead dividing 1 by that large value. Hence, the larger the positive exponent, the smaller the decimal result of the reciprocal will be, because you are dividing 1 by a larger number.