Problem 4 - Technical Mathematics with Calculus

Question

Write each power of 10 as a decimal number. $$10^{-1}$$

Step-by-Step Solution

Answer

The power of 10 for \(10^{-1}\) as a decimal number is 0.1.

Step 1: Understanding Negative Exponents

A negative exponent indicates one over the corresponding positive exponent. The expression \(10^{-1}\) tells you to take the reciprocal of \(10\) raised to the positive exponent 1.

Step 2: Converting to a Decimal

To convert \(10^{-1}\) to a decimal, determine the value of \(10\) to the first power and then find its reciprocal: \(10^1 = 10\), so \(10^{-1} = \frac{1}{10}\).

Step 3: Expressing the Reciprocal as a Decimal

The reciprocal of 10, or \(\frac{1}{10}\), is equal to 0.1 when expressed as a decimal number.

Key Concepts

Understanding Negative Exponents

When you come across an expression with a negative exponent, such as \(10^{-1}\), it's an indication to perform a mathematical inversion of the base number raised to the corresponding positive exponent. Essentially, a negative exponent reveals how many times you need to divide 1 by the base number. For example, \(10^{-1}\) equates to dividing 1 by 10, or the reciprocal of 10 raised to the first power.

In general, any number raised to a negative exponent \( x^{-n} \) can be rewritten as \( \frac{1}{x^n} \), where \( x \) is the base and \( n \) is the positive exponent. This conversion simplifies the process of working with negative exponents by transforming them into positive exponents nested within a fraction.

Decimal Conversion

Converting powers of 10 with negative exponents into decimal form involves a two-step process: finding the reciprocal of the base 10 to the positive power and then expressing that reciprocal as a decimal. Taking our example, \(10^{-1}\) is the same as \( \frac{1}{10} \), which can be understood as one whole divided into ten equal parts. In this context, each part is 0.1 of the whole.

To express a reciprocal as a decimal, identify how many zeroes should follow the decimal point. The number of zeroes is determined by the positive exponent after converting the negative one: if the exponent is 1 (as in \(10^{-1}\)), there will be one zero to the right of the decimal point - this leads us to 0.1. If it was \(10^{-2}\), there would be two zeros, leading us to 0.01, and so on, making the decimal conversion of powers of 10 with negative exponents a systematic process.

Reciprocal of a Number

The reciprocal of a number is what you get when you divide 1 by that number. Reciprocals are critical when dealing with negative exponents and provide a way to transform division into multiplication, which is often easier to handle. To find the reciprocal of any non-zero number \( x \), you simply calculate \( \frac{1}{x} \).

For instance, the reciprocal of 10 is \( \frac{1}{10} \) as previously discussed. Reciprocals are also relevant in fraction division – to divide by a fraction, you multiply by its reciprocal. In decimal terms, the process of finding the reciprocal may require understanding place value to ensure accurate conversion, especially for numbers not equal to powers of 10. It's a fundamental concept that underscores the operation of negative exponents and is a cornerstone in algebraic manipulation.