Problem 52 - College Algebra with Modeling and Visualization

Question

Write the number in standard form. $$ 0.0032 \times 10^{-1} $$

Step-by-Step Solution

Answer

The standard form is 0.00032.

Step 1: Understand the Problem

We need to represent the number \(0.0032 \times 10^{-1}\) in standard form, which involves writing it as a single decimal number.

Step 2: Evaluate the Power of Ten

The given expression is \(0.0032 \times 10^{-1}\). The power of ten, \(10^{-1}\), indicates that we need to divide 0.0032 by 10. This means we will move the decimal point in 0.0032 one place to the left.

Step 3: Move the Decimal Point

Moving the decimal point one place to the left in 0.0032, we change it to 0.00032. This movement represents the multiplication by \(10^{-1}\).

Step 4: Write the Final Answer

The result after moving the decimal point is 0.00032. This is the standard form of the initial expression \(0.0032 \times 10^{-1}\).

Key Concepts

Power of Ten

Powers of ten are a fundamental concept in mathematics that help express numbers in a more simplified way. When we talk about the power of ten, we're referring to how many times we multiply 10 itself. For instance, when we write \(10^2\), it means \(10 \times 10 = 100\). It's an easy way to shorthand larger numbers.

When dealing with powers of ten, it's important to understand whether they have a positive or negative exponent.

• A positive exponent means to multiply. For example, \(10^3 = 1000\).
• A negative exponent indicates division or multiplication by a fraction. For instance, \(10^{-1}\) means we divide by 10, or equivalently, multiply by \(\frac{1}{10}\).

In the context of the exercise, the power of ten is \(10^{-1}\). This tells us to move the decimal in the given number one place to the left to effectively divide it by 10. This movement adjusts the value, making it smaller, by shifting each digit one place further to the right.

Decimal Numbers

Decimal numbers are numbers that have a point separating the whole part and the fractional part. They are incredibly useful for representing values that are not whole numbers. Decimal numbers are based on the system of base 10, which is why they function so nicely with powers of ten.

When you see a decimal like 0.0032, the numbers to the right of the decimal point indicate fractions that are related to powers of ten. In this case:

• The first 3 indicates 3 tenths of a thousands, or \(3 \times \frac{1}{1000}\).
• The following number 2 is in the ten-thousandths place, representing \(2 \times \frac{1}{10000}\).

Operations with decimals often require moving the decimal point. By knowing how to manipulate this point, like in the exercise provided, you can convert them to standard form efficiently. Standard form takes decimal numbers and expresses them cleanly for easier calculation or understanding.

Mathematical Notation

Mathematical notation is a set of symbols used to represent numbers and operations in a concise way. This shorthand not only makes it easier to communicate mathematical ideas but also simplifies solving complex problems.

Standard form is part of mathematical notation. It allows us to express complex operations, like multiplication by powers of ten, succinctly. In the exercise, \(0.0032 \times 10^{-1}\) is simplified using this notation to represent as \(0.00032\).

• When you multiply by \(10^{-1}\), it translates to moving the decimal left by one position.
• Mathematical notation avoids excessive detail and focuses on the essential operation. This helps with clarity and prevents confusing lengthy descriptions.

Understanding mathematical notation involves familiarization with its conventions, like how we read decimal points and powers of ten. Once you get the hang of it, analyzing and solving problems becomes more intuitive and efficient.