Problem 5 - Technical Mathematics with Calculus

Question

Write each power of 10 as a decimal number. $$10^{4}$$

Step-by-Step Solution

Answer

The power of 10 to the 4th, written as a decimal, is 10,000.

Step 1: Understanding Powers of 10

Writing a power of 10 as a decimal involves understanding that the exponent tells us how many times the number 10 is multiplied by itself. The number 10 raised to an exponent is simply a 1 followed by that many zeros.

Step 2: Writing the Power as a Decimal

Since we have the power of 10 to the 4th power, this means we have four zeros after the 1. In decimal form, this is written as 10000.

Key Concepts

Exponents

To understand exponents, imagine you have a pile of cookies, and you are allowed to double the pile a certain number of times. If you double the pile once, you have twice as many cookies. If you double it twice, you've quadrupled the original pile. Exponents work similarly, but rather than doubling, we're 'multiplying by 10' a specific number of times indicated by the exponent.

Let's take the number 10 raised to the power of 4, or in mathematical terms, written as \(10^{4}\). This exponent, 4, directs us to multiply 10 by itself four times: \(10 \times 10 \times 10 \times 10\), which equals 10,000. In essence, an exponent notes how many times you use 10 as a multiplier.

Decimal Notation

Decimal notation is a way of writing numbers that includes a decimal point to signify the fractional part of the number. It follows the base-10 system, which is why it's also called 'base-10' or 'denary' system. Each position in a decimal number has a value that is a power of 10, based on its location relative to the decimal point.

In the exercise with \(10^{4}\), we have a whole number without a fractional part. The decimal notation of \(10^{4}\) is 10,000. Here, no decimal point is visible because there are no fractions involved. Looking to the right of the decimal for tenths, hundredths, and so on—you will find zeros or the absence of any number, indicating a whole number.

Scientific Notation

Scientific notation is a method to write very large or very small numbers succinctly. It is frequently used in science to handle the wide range of values that can occur. In scientific notation, numbers are written as a product of two factors: a coefficient that is a number between 1 and 10, and a power of 10.

For example, \(10^{4}\) in scientific notation is \(1 \times 10^{4}\). This format is particularly handy when dealing with much larger or smaller exponents, as it simplifies understanding and calculation. By stating \(1 \times 10^{4}\), it immediately indicates a value of 10,000, painting a clearer picture of its magnitude.