Problem 37 - College Algebra with Modeling and Visualization

Question

Write the number in scientific notation. $$ -0.0087 $$

Step-by-Step Solution

Answer

The number \(-0.0087\) in scientific notation is \(-8.7 \times 10^{-3}\).

Step 1: Identify the Decimal Part

First, locate the first nonzero digit in the number \(-0.0087\). The first nonzero digit is 8.

Step 2: Position the Decimal Point

Move the decimal point to the right of the first nonzero digit. In \(-0.0087\), moving the decimal point after the 8 gives you \(-8.7\). You moved the decimal point 3 places to the right.

Step 3: Determine the Exponent

Since the decimal point was moved 3 places to the right, you will use an exponent of \(-3\). In scientific notation, this movement is represented as a power of 10. So, you have \(-8.7 \times 10^{-3}\).

Step 4: Write in Scientific Notation

Combine the results from the previous steps. The number \(-0.0087\) in scientific notation is \(-8.7 \times 10^{-3}\).

Key Concepts

Decimal Numbers

Decimal numbers are a way of expressing fractions in a more readable form, using a dot to separate the whole numbers from the fractional parts. They are called 'decimal' because they are based on the power of ten. For example, the number 0.5 is a decimal number, which represents the fraction \( \frac{1}{2} \).

Decimal numbers allow for more accurate and simplified calculations in our day-to-day mathematical interactions. Here are some features of decimal numbers:

• The position of each number after the decimal point represents a power of 10 (tenths, hundredths, thousandths, etc.).
• They are often used to denote small or precise values, making them useful in scientific contexts.
• In cases like \(-0.0087\), the number becomes challenging to interpret at a glance, making it a perfect candidate for scientific notation.

Understanding decimal numbers is key to converting them into other formats like scientific notation, where they often become simpler to read and use.

Powers of 10

Powers of 10 play a crucial role when working with scientific notation. When we talk about powers of 10, we're talking about expressions like \( 10^2 \), \( 10^3 \), or \( 10^{-3} \). These tell us how many times to multiply or divide the number 10.

Here's how powers of 10 work in a practical sense:

• Positive exponents, like \( 10^3 \), mean to multiply 10 by itself for the number of times indicated (i.e., \( 10 imes 10 imes 10 = 1000 \)).
• Negative exponents, such as \( 10^{-3} \), mean to divide 1 by 10 raised to that power (i.e., \( \frac{1}{10^3} = 0.001 \)).
• Moving the decimal point to the right in a number involves using negative exponents because you're dividing by a power of ten, whereas moving to the left requires a positive exponent.

In scientific notation, using powers of 10 allows us to write large or small numbers efficiently by encoding the number of places the decimal point moves in the exponent.

Exponents

Exponents are a mathematical way of expressing repeated multiplication. The concept of exponents is fundamental in scientific notation, where you see them utilized as powers of ten. When you write a number as \( 8.7 \times 10^{-3} \), the exponent \(-3\) shows the power of 10 the number is being divided by.

Here are some important facts about exponents:

• An exponent tells you how many times to multiply a base number by itself. For example, \(5^3\) means \(5 \times 5 \times 5\), which equals 125.
• If the exponent is negative, it shows how many times the base should be divided, indicating that the value is less than one.
• Exponents simplify the representation of very small or very large numbers, making calculations faster and reducing possible errors.

In scientific notation, exponents are vital as they quickly convey the size of the number, whether it's a giant galaxy or something as minuscule as an atom.