Problem 2 - Technical Mathematics with Calculus

Question

Write each power of 10 as a decimal number. $$10^{-2}$$

Step-by-Step Solution

Answer

\(10^{-2}\) equals 0.01.

Step 1: Understanding Negative Exponents

A negative exponent indicates that the number is the reciprocal of the base raised to the absolute value of the exponent. In other words, to calculate a number with a negative exponent, you take the reciprocal of the base number raised to the positive exponent. For the base 10, this means that we will be dividing 1 by 10 raised to the power of the positive exponent.

Step 2: Convert the Exponent to Positive

To find the value of \(10^{-2}\), we first convert the negative exponent to a positive exponent by finding the reciprocal. The reciprocal of \(10^{2}\) is \(1/10^{2}\).

Step 3: Calculate the Positive Exponent

Next, we calculate \(10^{2}\) which equals \(10 \times 10 = 100\).

Step 4: Apply the Reciprocal

Now, we take the reciprocal of 100, which is \(1/100\), to find the decimal value of \(10^{-2}\).

Step 5: Convert to Decimal

Finally, we convert the fraction \(1/100\) to a decimal by dividing the numerator by the denominator. This equals 0.01.

Key Concepts

Reciprocal of Base Number

When dealing with negative exponents, the concept of a 'reciprocal' is key to understanding how to manipulate and evaluate expressions. A reciprocal is essentially a mathematical term for flipping a fraction. For any number that isn't zero, its reciprocal is found by dividing one by that number. If the number in question is a whole number, like our base number 10, we consider it as a fraction with 1 as the numerator and the number itself as the denominator; thus, 10 is written as \(\frac{1}{10}\).

Here's what happens with negative exponents: The negative sign prompts us to take the reciprocal of the base number. So, when we have an expression like \(10^{-2}\), we're essentially looking at it as \(\frac{1}{10^2}\) - we've taken the base number 10, turned it into a reciprocal (\(\frac{1}{10}\)), and then raised it to the power of 2. Remember, the reciprocal of a number multiplied by itself—like \(\frac{1}{10}\) times \(\frac{1}{10}\)—leads to \(\frac{1}{100}\), which is a fraction that we can convert into a decimal to simplify.

Converting Exponents

Converting exponents from negative to positive is a useful skill in mathematics that simplifies the process of finding the value of numbers with negative exponents. The trick lies in understanding that a negative exponent means the base will behave inversely.

Take for instance \(10^{-2}\). To convert the exponent from negative to positive, you must find the reciprocal of the base raised to the positive exponent. Thus, \(10^{-2}\) becomes \(1/10^2\), which we know is the same as \(1/100\) when we carry out the exponentiation. In simpler terms, converting the negative exponent to a positive one flips the 'position' of the base number - instead of multiplying 10 twice, we now divide 1 by 10 twice. The essence is to ensure that we're working with a positive exponent that's easier to evaluate, since our brains naturally understand multiplication more than the concept of division by a power.

Decimal Conversion

The last step in understanding powers of 10 with negative exponents involves converting fractions into decimal numbers. Decimal conversion helps us to represent numbers in a way that is often more understandable and tangible, especially when dealing with fractions.

To convert a fraction like \(1/100\) into a decimal, you divide the numerator (the top number) by the denominator (the bottom number). So for \(1/100\), you divide 1 by 100, which equals 0.01. This process converts our fraction into a decimal number, giving us a clear picture of the size of the number relative to 1. For exponents with larger negative values, such as \(10^{-3}\), which is \(1/1000\), dividing 1 by 1000 gives us 0.001. The pattern here is clear: for each power of 10 with a negative exponent, we add another zero to the left of 1 after the decimal point, shifting its value increasingly smaller.