Problem 29 - Fundamentals of Mathematics

Question

Expand the terms. (Do not find the actual value.) \(30^{2}\)

Step-by-Step Solution

Answer

\(30^2\) expands to \(400 + 400 + 100\).

Step 1: Identify the property to use

To expand the term \(30^2\), we will use the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\). This identity is used to expand terms of the form \(a+b\) squared.

Step 2: Express 30 as a sum

Write 30 as a sum of two numbers. For this exercise, we can express 30 as \(30 = 20 + 10\), where \(a = 20\) and \(b = 10\).

Step 3: Apply the identity

Using the identity \((a+b)^2 = a^2 + 2ab + b^2\), substitute \(a = 20\) and \(b = 10\). This gives us:\[ (20 + 10)^2 = 20^2 + 2 \cdot 20 \cdot 10 + 10^2 \]

Step 4: Simplify each term

Let's calculate each part of the expansion separately:- \(20^2 = 400\)- \(2 \cdot 20 \cdot 10 = 400\)- \(10^2 = 100\)

Key Concepts

Expanding Expressions

Expanding expressions is an essential algebraic process that involves breaking down an expression into simpler parts to make computation or further manipulation easier. In algebra, particularly with polynomials, expanding expressions transforms a compact form into an extended form.

For example, algebraic identities such as \((a + b)^2 = a^2 + 2ab + b^2\) are commonly used to expand binomial expressions. The goal is to express the given terms explicitly in terms of their individual parts.

• Application: It's particularly useful in solving equations and simplifying expressions.
• Benefits: Expanding helps in visualizing the components of an expression, making it easier to add, subtract, and perform other operations.
• Examples: Converting expressions like \((x+1)^2\) into expanded form: \(x^2 + 2x + 1\).

To master this technique, practice is key. Familiarity with algebraic identities will speed up the process.

Squaring Binomials

Squaring binomials is a specific case of expanding expressions where you multiply a binomial by itself. This involves using a special algebraic identity: \((a+b)^2 = a^2 + 2ab + b^2\).

This identity provides a systematic way to expand the squared terms.
To illustrate, consider the expression \( (20 + 10)^2 \). Here, the terms are recognized as a binomial, with \( a = 20 \) and \( b = 10 \).

Steps in Squaring Binomials

- Identify: Recognize the binomial form \((a+b)\) and identify the values of \(a\) and \(b\).
- Apply Identity: Use the identity to expand: \(a^2 + 2ab + b^2\).
- Compute: Calculate each component separately:

• \(20^2 = 400\)
• \(2 \cdot 20 \cdot 10 = 400\)
• \(10^2 = 100\)

The expanded form provides a clearer view of what the square of a binomial entails. This method simplifies complex calculations and helps in deeper understanding of polynomial changes.

Mathematical Properties

Understanding mathematical properties is vital in expanding and manipulating expressions in algebra. These properties govern how numbers and operations behave, providing consistency and predictability in calculations.

In the context of expanding expressions, properties like the Distributive Property play a crucial role.

Key Properties Used in Algebra

• Distributive Property: This property states that \(a(b + c) = ab + ac\). It's fundamental in expanding expressions as it allows you to distribute multiplication over addition.
• Commutative Property: This property signifies that order does not matter in addition or multiplication: \(a + b = b + a\) and \(ab = ba\).
• Associative Property: This property lets you regroup numbers for addition or multiplication: \((a + b) + c = a + (b + c)\).

By applying these properties, algebraic expansion becomes a seamless process that is both logical and efficient. In the example of \(30^2\), these properties were used implicitly to rearrange and simplify the expression step-by-step. Mastery of these properties enhances algebraic thinking and problem-solving skills.