Problem 37 - Algebra 1

Question

Simplify the expression. $$(\sqrt{a}-b)^{2}$$

Step-by-Step Solution

Answer

The simplified form of the expression \((\sqrt{a}-b)^{2}\) is \(a - 2 * \sqrt{a} * b + b^{2}\).

Step 1: Understanding the problem

The expression \((\sqrt{a}-b)^{2}\) needs to be simplified. This is equivalent to the square of the binomial expression \(\sqrt{a}-b\). Use the expansion of \((a-b)^{2}\) to simplify.

Step 2: Apply the formula of square of binomial

The square of a difference, or \((a-b)^{2}\), is equal to \(a^{2}-2ab+b^{2}\). Applying this formula to our expression where \(a = \sqrt{a}\) and \(b = b\), we get \((\sqrt{a})^{2}-2*\sqrt{a}*b+b^{2}\).

Step 3: Simplify further

The \((\sqrt{a})^{2}\) leads to \(a\), because the square root and the square operations cancel each other out. So, the result is \(a - 2 * \sqrt{a} * b + b^{2}\).

Key Concepts

Understanding Binomial Expansion

Binomial expansion is a method used to expand expressions that involve the power of a binomial. A binomial is an expression that contains two terms, like \((x + y)\). The expansion follows a specific pattern based on the power to which the binomial is raised. For instance, the expansion of \((a - b)^2\) is \(a^2 - 2ab + b^2\).
This pattern allows us to quickly expand the expression without multiplying the terms directly. It relies on recognizing the structure of the binomial and applying the formula accordingly. When simplifying, notice common patterns and use the right formula to make the process easy.

Exploring the Square of a Difference

The square of a difference uses the formula \((a - b)^2 = a^2 - 2ab + b^2\). This formula is essential because it tells us how to deal with expressions where two terms are subtracted and then squared.

• The first term, \(a^2\), is the square of the first component.
• The second term, \(-2ab\), is twice the product of both terms.
• The third term, \(b^2\), is the square of the second component.

This method simplifies expressions like \((\sqrt{a} - b)^2\) by allowing us to follow clear steps without getting lost in complex multiplication.

Steps in Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form. This process eliminates unnecessary complexity by combining like terms and using algebraic identities.
For the expression \((\sqrt{a} - b)^2\):

• Recognize patterns: Notice that it's in the form of a binomial squared.
• Apply known formulas: Use the \((a - b)^2 = a^2 - 2ab + b^2\) formula.
• Simplify components: Realize that \((\sqrt{a})^2 = a\), simplifying the expression further to \(a - 2\sqrt{a}b + b^2\).

Each step contributes to making the expression easier to work with and understand.

Introduction to Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. They come in various forms and can be simple or very complex.
Consider the polynomial \(a - 2\sqrt{a}b + b^2\) derived from our simplification. Here, each term:

• \(a\) is a simple term.
• \(-2\sqrt{a}b\) involves a product of a constant with a square root and a variable.
• \(b^2\) is a basic square term.

Understanding polynomials helps in breaking down expressions into manageable parts, allowing for easier manipulation and problem-solving.