Problem 82 - Algebra A Combined Function

Question

Simplify, if possible. $$ x^{5} y^{2}+y^{2} x^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^5y^2\).

Step 1: Identify Like Terms

To simplify the expression, observe if there are any like terms that can be combined. Notice that the terms \(x^5y^2\) and \(y^2x^5\) are actually the same because multiplication is commutative.

Step 2: Combine Like Terms

Since \(x^5y^2\) and \(y^2x^5\) are identical terms, we can add them together. Thus, the expression becomes \(x^5y^2 + x^5y^2\).

Step 3: Simplify the Expression

Combine the coefficients of the like terms. The simplified expression is \(2x^5y^2\).

Key Concepts

Understanding Like Terms

When trying to simplify algebraic expressions, identifying like terms is crucial. Like terms are terms that have the same variables raised to the same powers. It doesn't matter what their coefficients are, as long as everything else is identical. By identifying like terms, you can simplify your expressions more easily.
For example, in the expression \(x^5y^2\) and \(y^2x^5\):

• Both terms have the same variables \(x\) and \(y\).
• Both \(x\) and \(y\) are raised to the powers of 5 and 2, respectively.

This means they are like terms, making them perfect candidates for simplification.

The Commutative Property in Action

The commutative property is a basic principle of arithmetic that says you can swap numbers around and still get the same result when you add or multiply them. For example, \(a + b = b + a\) and \(ab = ba\). This property is incredibly useful in algebra because it allows you to rearrange terms to identify and combine like terms more easily.
In our exercise, the expressions \(x^5y^2\) and \(y^2x^5\) seem different at first glance. But, once we apply the commutative property, we see they are actually the same. This helps us to combine them effortlessly.
By recognizing the commutative property, you can simplify expressions without getting tripped up by the order of terms.

Mastering Combining Coefficients

Combining coefficients is the final step in simplifying an expression once you've identified the like terms. A coefficient is a number that multiplies a variable in an algebraic expression. To combine coefficients, you simply add or subtract them according to the arithmetic sign.
In our example, after recognizing that \(x^5y^2\) and \(y^2x^5\) are like terms, we had the expression \(x^5y^2 + x^5y^2\). The coefficient of \(x^5y^2\) is 1, because \(1 \, \cdot \, x^5y^2 = x^5y^2\). So, you simply add these coefficients:

• \(1x^5y^2 + 1x^5y^2 = 2x^5y^2\)

Thus, the final simplified form of the expression is \(2x^5y^2\). Combining coefficients smoothens out the expression and provides its simplest form.