Problem 122 Simplify. $$ \left(a^{2}+b^{... [FREE SOLUTION]

Chapter 1: Problem 122

Simplify. $$ \left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right) $$

Short Answer

Expert verified

The expression simplifies to $a^4 - b^4$.

Step by step solution

Identify the Formula

Recognize that the given expression $ (a^2+b^2)(a^2-b^2) $ is in the form of the difference of squares formula. The difference of squares formula is $ (x+y)(x-y) = x^2 - y^2 $. Here, let $ x = a^2 $ and $ y = b^2 $.

Apply the Formula

Apply the difference of squares formula to the expression. Substitute $ x = a^2 $ and $ y = b^2 $ into $ x^2 - y^2 $:\[(a^2+b^2)(a^2-b^2) = (a^2)^2 - (b^2)^2\]

Simplify the Expression

Simplify the expression obtained in the previous step. Calculate each squared term:\[(a^2)^2 = a^4 \quad \text{and} \quad (b^2)^2 = b^4\]Thus, the expression simplifies to:\[a^4 - b^4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Identities

Polynomial identities are special algebraic expressions that reveal relationships between different polynomials. These identities showcase equalities that hold true for all values of the variables involved. Understanding these can help simplify complex expressions. In the given exercise, we use an important polynomial identity known as the difference of squares. This identity follows the pattern: $(x+y)(x-y) = x^2 - y^2$. By applying this, we can transform expressions into a more manageable form. Recognizing these identities is key in algebra as they simplify solving and working with variables.

Simplification Techniques

Simplification techniques make complex algebraic expressions easier to handle. In algebra, breaking down expressions into simpler parts helps in understanding and finding solutions quicker.

Identify patterns or identities that relate to the given expression, such as the difference of squares.
Substitute known identities into the expression to reduce its form. In this exercise, we apply the formula $(x^2 - y^2)$.
Calculate straightforward operations like squaring terms: $(a^2)^2$ becomes $a^4$.

These techniques are essential for achieving a clearer and simpler expression, enabling easier computation and understanding.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operators. They do not contain an "equals" sign, differentiating them from equations. The expression given in the exercise is composed of powers and variables: $ (a^2+b^2)(a^2-b^2)$.
Understanding algebraic expressions involves several key concepts:

*Variables*: Symbols like $a$ and $b$ that represent unknown values.
*Constants*: Numbers that have fixed values, although none are explicitly present here.
*Operations*: Additions, subtractions, multiplications, and exponents are found in the expression.

By mastering the structure and elements of algebraic expressions, one can manipulate and simplify them effectively, leading to more advanced problem-solving skills.

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91影视

Simplify. $$ \left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right) $$

Short Answer

Step by step solution

Identify the Formula

Apply the Formula

Simplify the Expression

Key Concepts

Polynomial Identities

Simplification Techniques

Algebraic Expressions

One App. One Place for Learning.

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