Problem 252 Find each product. $$ \left(... [FREE SOLUTION]

Chapter 5: Problem 252

Find each product. $$ \left(x^{5}+y^{5}\right)\left(x^{5}-y^{5}\right) $$

Short Answer

Expert verified

\[ x^{10} - y^{10} \]

Step by step solution

Identify the Formula

Recognize that the given expression $\text{(x^5 + y^5)(x^5 - y^5)}$ can be simplified using the difference of squares formula, which states: \[a^2 - b^2 = (a + b)(a - b) \]

Apply the Formula

In this context, let $a = x^5$ and $b = y^5$. Substitute these into the general formula: \[ (x^5)^2 - (y^5)^2 \]

Simplify the Expression

Simplify to $(x^5)^2 - (y^5)^2 = x^{10} - y^{10}$

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Key Concepts

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

Algebraic formulas

In algebra, there are specific formulas that help simplify and solve expressions. One key formula is the 'Difference of Squares.' This formula, written as $a^2 - b^2 = (a + b)(a - b)$, indicates that the product of a sum and difference of two terms results in the difference of their squares. Recognizing and applying such formulas can make solving polynomial expressions much easier. By identifying similar patterns in expressions, we can transform them into simpler forms for easier computation. This is like having a shortcut that saves time and effort in algebraic procedures.

Polynomial expressions

Polynomial expressions involve sums and differences of terms with variables raised to various powers. In the given exercise, $x^5 + y^5$ and $x^5 - y^5$ are both polynomial expressions. A polynomial could have any number of terms, but the important thing about polynomials is how they behave and how we can manipulate them using algebraic rules. When multiplying polynomials, it is helpful to look for ways to use formulas to simplify the process, like using the difference of squares formula in our problem. Understanding the structure and properties of polynomials is essential in advanced algebra and calculus.

Exponent rules

Exponents are fundamental in algebra and any higher-level math. In the exercise, we saw terms like $x^5$ and $y^5$. Remember, an exponent indicates how many times a number is multiplied by itself. For example, $x^5$ means $x \times x \times x \times x \times x$. The rules of exponents are crucial when simplifying expressions. One key rule used in the solution was multiplying exponents: $(x^5)^2 = x^{10}$. This rule states that when you raise a power to another power, you multiply the exponents together. Mastering these rules ensures a smoother algebraic manipulation and problem solving.

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91影视

Find each product. $$ \left(x^{5}+y^{5}\right)\left(x^{5}-y^{5}\right) $$

Short Answer

Step by step solution

Identify the Formula

Apply the Formula

Simplify the Expression

Key Concepts

Algebraic formulas

Polynomial expressions

Exponent rules

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