91Ó°ÊÓ

Combining like terms is a fundamental process in algebra that helps simplify expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(2x + 3x - x\), all three terms are like terms because they all involve the variable \(x\). To combine them, you simply add or subtract their coefficients. In this case, \(2x + 3x - x = 4x\).
For our exercise, the given expression is: \[ t^{2}-5w-(-2w-t^{2}) \].
We first need to identify the like terms. Here, \(t^{2}\) and \(-t^{2}\) are like terms, and \(-5w\) and \(-2w\) are like terms. Then, combine them to simplify the expression by adding or subtracting their coefficients: \((t^{2} + -t^{2} = 0)\) and \((-5w + 2w = -3w)\).
So, our simplified terms are \[0 - 3w \rightarrow -3w \].
In short, combining like terms makes complex expressions simpler and easier to work with.

Distributing a negative sign involves changing the signs of the terms within parentheses. This is crucial when simplifying expressions. In our exercise, we have the negative sign before the parentheses: \[ -( -2w - t^{2}) \].
To distribute the negative sign, change the signs of all the terms inside the parentheses:

• The \(-2w\) becomes \[ -(-2w) = 2w \].
• The \(-t^{2}\) becomes \[ -(-t^{2}) = t^{2} \].

Therefore, the expression \[-(-2w - t^{2})\] becomes \[2w + t^{2}\]. Distributed negative signs can also appear in more complex forms but the fundamental principle remains the same.

Algebraic simplification involves performing operations to rewrite an expression in the simplest form. It includes distributing signs, combining like terms, and performing arithmetic operations.
Let's apply these steps to our exercise. The original expression is: \[ t^{2}-5w-(-2w-t^{2}) \].
Following the distributive property, we first distribute the negative sign:

• Distribute negative sign: \[ t^{2}-5w+2w+t^{2} \].

After distributing the negative sign, we combine the like terms:

• Combine like terms: \[ t^{2}+t^{2} = 2t^{2} \] and \[ -5w + 2w = -3w \].

Finally, by simplifying the given expression, we arrive at:

• Replacing the like terms with their coefficients: \[-3w+2t^{2}\rightarrow 2t^{2}-3w \]

Hence, the simplest form of the expression \[t^{2}-5w-(-2w-t^{2}) \] is \[2t^{2} - 3w \]. Achieving simplification makes algebraic expressions easier to understand and solve.