Problem 79 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \(\left(x^{2}+y^{2}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^4 + 2x^2y^2 + y^4\).

Step 1: Recognize the Expression Structure

The given expression is \((x^2 + y^2)^2\), which is a binomial expression raised to the second power. Our task is to expand it using the formula \((a+b)^2 = a^2 + 2ab + b^2\).

Step 2: Substitute Values into the Formula

Identify \(a = x^2\) and \(b = y^2\). Substitute these values into the expansion formula: \((x^2)^2 + 2(x^2)(y^2) + (y^2)^2\).

Step 3: Simplify Each Term

Calculate each term separately: - \((x^2)^2 = x^4\)- \(2(x^2)(y^2) = 2x^2y^2\)- \((y^2)^2 = y^4\)

Step 4: Combine the Simplified Terms

Write out the simplified terms together to get the expression: \(x^4 + 2x^2y^2 + y^4\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool used in algebra for expanding expressions that are raised to a power. In simple terms, it provides a method to simplify expressions like \((a + b)^n\). The Binomial Theorem states that:

• \((a + b)^n = \sum_{k=0}^{n} C(n, k) a^{n-k}b^{k}\)
• \(C(n, k)\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\)

For cases like \((x^2 + y^2)^2\), the formula \((a + b)^2 = a^2 + 2ab + b^2\) can be directly applied. You identify each component of the binomial and apply the rule to find the expanded form. Here, \(a = x^2\) and \(b = y^2\), leading to \(a^2 + 2ab + b^2 = x^4 + 2x^2y^2 + y^4\). This method efficiently breaks down complex expressions.

Algebraic Expressions

Algebraic expressions form the foundation of algebra. They are made up of variables, numbers, and operations. An algebraic expression can be as simple as \(2x + 3\) or more complex like \((x^2 + y^2)^2\). In an expression like this:

• Variables: Symbols that represent numbers, such as \(x\) and \(y\).
• Constants: Fixed numbers, which are 2 and 3 in the case of what's inside \((a + b)\).
• Operations: Such as addition, subtraction, multiplication, and division.

When working with algebraic expressions, the goal is often to expand or simplify them. By using rules such as the binomial theorem or distribution, these expressions become manageable. Expressions like the given \((x^2 + y^2)^2\) can be viewed as a challenge to interpret and simplify by identifying components and applying known algebraic rules.

Exponentiation

Exponentiation is a mathematical operation involving numbers or variables called bases raised to powers, referred to as exponents. In the expression \(a^n\), \(a\) is the base and \(n\) is the exponent, telling you how many times to multiply the base by itself. For example, \(x^2\) means \(x \times x\).When expanding expressions like \((x^2 + y^2)^2\), exponentiation is integral to understanding and simplifying the terms:

• In \((x^2)^2\), the base \(x^2\) is squared, leading to \(x^4\) with the rule \((a^m)^n = a^{m \cdot n}\).
• Similarly, \((y^2)^2\) becomes \(y^4\).
• The cross term \(2(x^2)(y^2)\) involves exponent rules as well, resulting in \(2x^2y^2\).

Understanding how to manipulate exponents is crucial for working effectively with algebraic expressions, ensuring accurate expansion and simplification. In our example, exponentiation allows breaking down each term of the binomial expansion accurately, resulting in a clear and concise outcome.