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The distributive property is a fundamental algebraic principle that allows you to multiply a single term across a sum or difference. In binomial expressions like \((2 \text{√}{x} - 3)(3 \text{√}{x} + 4)\), the property ensures each term in the first binomial is multiplied by every term in the second binomial.
In our example, it breaks down as follows:

• First terms: \((2\text{√}{x})(3\text{√}{x})\)
• Outside terms: \((2\text{√}{x})(4)\)
• Inside terms: \((-3)(3\text{√}{x})\)
• Last terms: \((-3)(4)\)

This essentially unpacks the expression into simpler parts, making it easier to manage and solve.

The FOIL Method is a specific application of the distributive property for multiplying two binomials. FOIL stands for:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms in the product.
• Inside: Multiply the inside terms.
• Last: Multiply the last terms in each binomial.

So for our binomial \((2\text{√}{x} - 3)(3\text{√}{x} + 4)\), applying the FOIL method gives you:

• First: \((2\text{√}{x})(3\text{√}{x})\)
• Outside: \((2\text{√}{x})(4)\)
• Inside: \((-3)(3\text{√}{x})\)
• Last: \((-3)(4)\)

Once we expand the expression using these steps, we can move on to simplifying each term further.

Combining like terms is a process of simplifying an expression by adding or subtracting the coefficients of terms that have the same variable and exponent. For our binomial product \((2 \text{√}{x} - 3)(3 \text{√}{x} + 4)\), after expansion, we get:
ewline \text{(First terms)} \((2\text{√}{x})(3\text{√}{x}) = 6x\)
\text{(Outside terms)} \((2\text{√}{x})(4) = 8\text{√}{x}\)
\text{(Inside terms)} \((-3)(3\text{√}{x}) = -9\text{√}{x}\)
\text{(Last terms)} \((-3)(4) = -12\).
Combining like terms means we group similar terms together:

• \(6x + 8\text{√}{x} - 9\text{√}{x} - 12\)

The like terms here are \(8\text{√}{x}\) and \(-9\text{√}{x}\), which sum up to \(-\text{√}{x}\). Therefore, the simplified expression is:

• \(6x - \text{√}{x} - 12\)

This makes it much clearer and easier to understand.

Radicals (also called roots) are expressions that involve a square root, cubic root, or higher-order root. In our original problem, we had radicals of \(\text{√}{x}\). When dealing with radicals:

• Multiplying two identical radicals: \(\text{√}{a} \times \text{√}{a} = a\)
• Simplify any radicals wherever possible.

So when we handled the radicals in the expanded form, such as \((2\text{√}{x})(3\text{√}{x})\), we simplified by noting that \(\text{√}{x} \times \text{√}{x} = x\), leading to an overall term of \(6x\). It's crucial to recognize and simplify radicals to combine like terms effectively and solve the expression.