Problem 12 - Elementary and Intermediate Algebra

Question

Express each power of 10 in fraction form and decimal form. a. \(10^{-3}\) b. \(10^{-6}\)

Step-by-Step Solution

Answer

10^{-3} is \(\frac{1}{1000}\) or 0.001, and 10^{-6} is \(\frac{1}{1,000,000}\) or 0.000001.

Step 1: Understanding Negative Powers of 10

Powers of 10 where the exponent is negative, such as \(10^{-n}\), means we take the reciprocal of \(10\) raised to the positive \(n\). Thus, \(10^{-n} = \frac{1}{10^n}\). We will use this understanding in the next steps.

Step 2: Fraction Form for \(10^{-3}\)

Using the rule from Step 1, \(10^{-3}\) in fraction form is \(\frac{1}{10^3}\). Calculate \(10^3 = 1000\), so \(10^{-3} = \frac{1}{1000}\).

Step 3: Decimal Form for \(10^{-3}\)

Convert the fraction \(\frac{1}{1000}\) into decimal form, which is 0.001. This is done by checking how many times 10 multiplies itself, i.e., 3 times in the denominator, resulting in placing three zeros after the decimal.

Step 4: Fraction Form for \(10^{-6}\)

Again applying the rule, \(10^{-6}\) in fraction form is \(\frac{1}{10^6}\). Calculate \(10^6 = 1,000,000\), so \(10^{-6} = \frac{1}{1,000,000}\).

Step 5: Decimal Form for \(10^{-6}\)

Translate \(\frac{1}{1,000,000}\) into decimal form. Since the fraction denominator has 6 zeros, the decimal representation is written as 0.000001.

Key Concepts

Fraction Form

When we talk about fraction form, we're looking at how to express negative powers of 10 as fractions. The main idea is that a negative exponent indicates taking the reciprocal (or inverse) of the number raised to the corresponding positive exponent. Therefore, the rule to remember is that any negative power of 10, like \(10^{-n}\), is equivalent to \(\frac{1}{10^n}\).
You encounter this when you need to represent a number like \(10^{-3}\) or \(10^{-6}\).

• For \(10^{-3}\), you'd convert it to fraction form as \(\frac{1}{10^3}\). Calculating further, since \(10^3 = 1000\), it becomes \(\frac{1}{1000}\).
• Similarly, for \(10^{-6}\), written as fraction \(\frac{1}{10^6}\), you find that \(10^6 = 1,000,000\) or one million. So, \(10^{-6}\) equals \(\frac{1}{1,000,000}\).

By changing negative powers of 10 into fractions, you're simply finding a practical way to represent very small numbers in a more intuitive form.

Decimal Form

Decimal form is a direct way to see how small a number is when dealing with negative powers of 10. After converting the number into fraction form, the next step is to translate it into decimal form. This is often more intuitive.

• To express \(10^{-3}\) in decimal form, note that it is equivalent to \(\frac{1}{1000}\). In decimal, this would be 0.001. The three zeros after the decimal point signify that the decimal point shifts to the left three places, making the number quite small.
• Similarly, \(10^{-6}\) as a fraction \(\frac{1}{1,000,000}\) becomes 0.000001 in decimal form. Here, six zeros follow the decimal point because the denominator of the fraction, 1,000,000, has six zeros.

Converting negative powers of 10 into decimal form can greatly simplify calculations, and makes it easier to visualize the scale of the number.

Exponents

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. In the case of powers of 10, exponents are extremely useful for representing very large or very small numbers.

• A positive exponent like \(10^n\) means 10 is multiplied by itself \(n\) times. For example, \(10^3 = 10 \times 10 \times 10 = 1000\).
• On the other hand, a negative exponent such as \(10^{-n}\) represents the reciprocal of the positive power. This means you switch the normally multiplied base into a fraction: \(\frac{1}{10^n}\).

Negative exponents help to relate to extremely small values by utilizing fractional representations instead of having to deal with cumbersome decimals. Whether positive or negative, exponents provide a convenient method to handle numbers with repetitive multiplicative patterns in either direction along the number line.