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Chapter 5: Problem 26

Simplify each expression. $$ x^{2} y^{3} x^{3} y^{2} $$

Short Answer

Expert verified
The expression simplifies to \( x^5 y^5 \).

Step by step solution

01

Identify like terms

In the expression given, identify like terms based on their variables. Here, the like terms are similar in nature and can be grouped based on the variables they are associated with: terms involving \( x \) are \( x^2 \) and \( x^3 \); terms involving \( y \) are \( y^3 \) and \( y^2 \).
02

Apply the property of exponents

Apply the property of exponents for multiplication: when multiplying terms with the same base, add their exponents. For the \( x \) terms, it becomes \( x^{2+3} = x^5 \). For the \( y \) terms, it becomes \( y^{3+2} = y^5 \).
03

Rearrange and simplify the expression

Combine both simplified terms to form the simplified expression. The expression turns into \( x^5 y^5 \), combining the results from the previous step.

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Key Concepts

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

Simplifying Algebraic Expressions
Simplifying algebraic expressions involves reducing them to their simplest form so that they are easier to work with, especially when solving equations. This process often includes:
  • Combining like terms
  • Applying the properties of exponents
  • Reorganizing terms to create a neater form
When simplifying, the goal is to make the expression as compact and understandable as possible. Expressions with multiple terms can be simplified by grouping like terms and then applying mathematical operations to reduce it to its smallest terms.
In the example, the expression \(x^2y^3x^3y^2\) becomes much easier to handle when simplified to \(x^5y^5\). This simplification aids in both clarity and calculation efficiency, especially in complex equations.
Like Terms
In algebra, 'like terms' are terms that have the same variables raised to the same power. Only the coefficients can differ. Recognizing like terms is crucial for simplifying expressions and solving equations easily.
Like terms can be:
  • Purely numerical, such as 3 and 5
  • Contain variables, such as \(2x\) and \(5x\)

In the original exercise, \(x^2\) and \(x^3\) are like terms because they both involve the variable \(x\). Similarly, \(y^3\) and \(y^2\) are also like terms. We only combine terms that are like to simplify the expression. Through like terms, we can apply the properties of exponents to merge them into one single term.
Algebra Basics
Understanding the basics of algebra is essential, as it forms the foundation for many branches of mathematics and real-world applications. Algebra involves using symbols, often standard letters, to represent numbers and quantities in formulas and equations.
This subject includes several key components:
  • Variables and constants, where variables can change and constants don't
  • Operations including addition, subtraction, multiplication, and division
  • The use of exponents to denote repeated multiplication
In the given example, the expression \(x^2y^3x^3y^2\) challenges us to really understand these algebra basics, particularly with exponent rules. By developing a strong foundation in these basics, you'll be better equipped to tackle more complicated problems and algebraic operations.

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Most popular questions from this chapter

Factor each expression. $$ x^{2}+20 x+100-9 z^{2} $$

Explain how the patterns used to factor the sum and difference of two cubes are similar and how they differ.

Find two trinomials such that their sum is a binomial and difference is a monomial.

Graph the solution set of each inequality or system of inequalities on a rectangular coordinate system. $$ x \geq 2 $$

What polynomial should be subtracted from \(5 x^{3} y-5 x y+2 x\) to obtain the polynomial \(8 x^{3} y-7 x y+11 x ?\)
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