Problem 47 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 21 x^{2} y^{3}+3 x y+x^{2} y^{3}+6 $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given algebraic expression is \(22x^2y^3 + 3xy + 6\).

Step 1: Identify like terms

In the given expression, $$ 21x^2y^3 + 3xy + x^2y^3 + 6, $$ we can see that \(21x^2y^3\) and \(x^2y^3\) are like terms because they have the same variables raised to the same powers.

Step 2: Combine like terms

To combine the like terms, we add their coefficients: $$ 21x^2y^3 + x^2y^3 = (21+1)x^2y^3 = 22x^2y^3 $$

Step 3: Rewrite the expression

Now, we rewrite the expression with the combined like terms: $$ 22x^2y^3 + 3xy + 6 $$ The simplified algebraic expression is $$ 22x^2y^3 + 3xy + 6. $$

Key Concepts

Combining Like Terms

Understanding how to combine like terms is a fundamental skill when working with algebraic expressions. Like terms are terms that have exactly the same variables raised to the same powers. The only difference between them would be the numerical coefficients.

For instance, in the expression
\[21x^2y^3 + 3xy + x^2y^3 + 6,\]
the terms \(21x^2y^3\) and \(x^2y^3\) are considered like terms because they share the same combination of variables, \(x^2\) and \(y^3\), indicating that their corresponding variables are raised to the same power. To combine them, simply add the coefficients (the numerical parts), and you get \(22x^2y^3\). Note that terms like \(3xy\) and \(6\) cannot be combined with these because the variables or their powers differ.

When simplifying expressions by combining like terms, always look for terms that match in variables and their exponents, add or subtract their coefficients, and keep the common variables unchanged.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Variables represent unknown values and are often denoted by letters such as \(x\), \(y\), and \(z\). Expressions can be as simple as \(3x + 4\) or as complex as the one in our exercise \(21x^2y^3 + x^2y^3 + 3xy + 6\).

In an expression, we use addition, subtraction, multiplication, and division to combine these variables and numbers. The goal when simplifying an expression is to make it as concise as possible while retaining its value. This involves combining like terms, as we've previously described, as well as applying other algebraic principles such as the distributive property or factoring.

Variables in Algebra

In algebra, variables are symbols that stand in for unknown numbers. They allow us to create general formulations or to state problems without specifying the exact values involved. In the expression we've looked at, \(x\) and \(y\) are variables. They can represent any number, and when combined with exponents, as in \(x^2\) or \(y^3\), they indicate that the variable is to be multiplied by itself a certain number of times.

Working with variables means understanding that they have a consistent value throughout an expression. For example, if \(x\) is 2 in one part of the expression, it must be 2 everywhere else in that expression. This consistency is the key to combining like terms successfully and to perform algebraic manipulations accurately.