Problem 25 - Thinking Mathematically

Question

Express each expanded form as a Hindu-Arabic numeral. \(\left(3 \times 10^{2}\right)+\left(8 \times 10^{1}\right)+(5 \times 1)\)

Step-by-Step Solution

Answer

The Hindu-Arabic numeral form of the given expression is 385.

Step 1: Simplify the Exponential Terms

For each term in the expression, calculate the product by raising 10 to the power specified and multiplying it by the coefficient. So, 1. \(3 \times 10^{2}\) equals \(3 \times 100\) which equals 300. 2. \(8 \times 10^{1}\) equals \(8 \times 10\) which equals 80. 3. \(5 \times 1\), which is 5.

Step 2: Sum the Products

Add the products from step 1 together. So, \(300 + 80 + 5 = 385\)

Key Concepts

Understanding Exponents

Exponents are a handy mathematical concept that help us understand repeated multiplication. The number, or base, is multiplied by itself as many times as the exponent indicates. For example, in the expression \(10^2\), the base is 10, and the exponent is 2. This means 10 is multiplied by itself, resulting in \(10 \times 10 = 100\). When expressing numbers in expanded form using exponents, each term shows how many times the base (usually 10 in the decimal system) is multiplied by itself.When you see an exponent in an expression, follow these steps:

• Identify the base and the power (exponent).
• Multiply the base by itself as many times as the exponent dictates.
• Multiply this result by the coefficient (the number in front).

For instance, in \(3 \times 10^2\), we calculate \(10^2\) first, getting 100. Then, multiply by 3 to get 300.

Expanded Form Basics

The expanded form is a method of breaking down a number to show its constituents based on place value. Instead of simply writing a number in its compact form, expanded form expresses the number as a sum of each digit multiplied by its corresponding place value.Consider the number 385. In expanded form, it can be expressed as:

• Hundreds: The digit 3, which actually means 300, as it is in the hundreds place.
• Tens: The digit 8, representing 80, found in the tens place.
• Units: The digit 5, which remains 5, located in the units place.

Writing numbers in expanded form using exponents, like \((3 \times 10^2) + (8 \times 10^1) + (5 \times 1)\), helps highlight the value of each part using powers of ten, making it clear how each digit contributes to the overall value of the number.

Place Value Importance

Place value is a fundamental concept in the decimal number system. It defines the position of a digit in a number and determines its actual value. The Hindu-Arabic numeral system is positional, meaning the value of a digit changes based on its position.Consider the number 385 again:

• The digit 5 is in the units place, carrying a value of 5.
• The digit 8 is in the tens place, giving it a value of 80 (\(8 \times 10\)).
• The digit 3 is in the hundreds place, which makes it 300 (\(3 \times 100\)).

Understanding place value is crucial when converting numbers from expanded form to their standard Hindu-Arabic numeral form. It guides the organization of digits into hundreds, tens, and units, or any other corresponding place for larger numbers. This concept not only helps in simplifying problems but also in appreciating the role of each digit's position in forming the full number.