Problem 20 - Algebra for College Students

Question

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((1.62)(10)^{2}\)

Step-by-Step Solution

Answer

The expression is 162 in decimal notation.

Step 1: Identify the Components

Recognize the given expression: \((1.62)(10)^{2}\). This consists of a base number, 1.62, and an exponent, 2, applied to 10.

Step 2: Calculate the Power of Ten

Compute \(10^{2}\), which refers to multiplying 10 by itself. So, \(10^{2} = 10 \times 10 = 100\).

Step 3: Multiply the Components

Multiply the base number by the power of ten calculated in the previous step: \(1.62 \times 100\).

Step 4: Convert into Ordinary Decimal Notation

Perform the multiplication: \(1.62 \times 100 = 162\). So, the expression \((1.62)(10)^{2}\) in ordinary decimal notation is 162.

Key Concepts

Scientific Notation

Scientific notation is a way of expressing really large or very small numbers conveniently. It's especially useful in science and engineering.
Using scientific notation, a number is written as the product of two parts:

• A decimal number greater than or equal to 1 but less than 10.
• An integer power of 10.

For example, the number 1.62 in scientific notation is written as \( (1.62)(10)^0 \).
When using scientific notation, you can easily see the size of the number and how many places the decimal point should move.If the power of 10 is positive, the decimal point moves to the right for each unit of increase in the exponent. If the exponent is negative, the decimal moves to the left. It’s a helpful technique for simplifying calculations and is essential for handling both extremely large and tiny numbers in various scientific fields.

Exponents

Exponents are a shorthand way to show how many times a number is multiplied by itself. In the expression \( (1.62)(10)^2 \), the number 10 is raised to the power of 2.
Here's how they work:

• The exponent indicates the number of times the base is multiplied by itself.
• For example, \( 10^2 \) means \( 10 \times 10 \) which is 100.

Exponents are crucial in scientific notation because they help to express large numbers concisely. For instance, instead of writing 1000, we can simply write \( 10^3 \). Exponents make multiplicative calculations more straightforward and reduce errors in manual computations. They play a significant role in various mathematical operations and help streamline complex equations.

Multiplication

Multiplication is one of the four basic operations in arithmetic; it involves finding the total of one number added repeatedly. When we're multiplying by powers of ten, the process becomes simplified, thanks to exponents.
Consider our problem: \( 1.62 \times 10^2 \) simplifies to \( 1.62 \times 100 \). Here’s why:

• Calculate the exponent first to get 100 (as \( 10^2 \)).
• Multiply 1.62 by 100, which gives you 162.

Multiplying by powers of ten is often just a matter of shifting the decimal point. For each increase by one in the power, move the decimal to the right by one place. This technique is very useful when converting numbers from scientific notation back to ordinary decimal form. Understanding these steps not only simplifies calculations but also strengthens your ability to work with various mathematical expressions.