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The distributive law is a key principle in algebra that allows us to multiply terms through addition or subtraction inside parentheses. It states that, for any real numbers \(a\), \(b\), and \(c\), the expression \(a(b - c)\) is equivalent to \(ab - ac\). This can be thought of as "distributing" the multiplication of \(a\) across both \(b\) and \(-c\).
In the given problem \((\sqrt{u}-3)(\sqrt{u}-4)\), we apply the distributive law. First, we distribute \(\sqrt{u}\) across both terms in the second binomial:

• \(\sqrt{u} \cdot \sqrt{u} = u\)
• \(\sqrt{u} \cdot -4 = -4\sqrt{u}\)

Then, we distribute \(-3\) across the terms in the second binomial:

• \(-3 \cdot \sqrt{u} = -3\sqrt{u}\)
• \(-3 \cdot -4 = 12\)

After distribution, the expression becomes \(u - 4\sqrt{u} - 3\sqrt{u} + 12\). This step sets the stage for further simplification.

Simplifying radicals is an important concept to understand in order to make expressions easier to work with. When we simplify the expression \(\sqrt{u}(\sqrt{u}-4)\), we deal with multiplying square roots. Specifically, multiplying \(\sqrt{u}\) by \(\sqrt{u}\) results in \(u\), because \((\sqrt{u})^2 = u\).
This operation follows from the property that the square of a square root cancels out the radical, leaving the term under the root as it is.
Another aspect of simplifying radicals involves ensuring all like radicals are reduced or combined later, as seen in this problem where \(-4\sqrt{u}\) and \(-3\sqrt{u}\) are handled.) Here, there are no further simplifications needed within \(-4\sqrt{u}\) or \(-3\sqrt{u}\) other than preparing them for the final combination.

Combining like terms is a fundamental process to simplify expressions further. Like terms are terms that have the exact same variable and the same exponent; they can be summed up by adding or subtracting their coefficients.
In our problem, the like terms are \(-4\sqrt{u}\) and \(-3\sqrt{u}\) because they share the same radical term \(\sqrt{u}\). By combining these terms, you simplify the expression by performing the arithmetic operation on their coefficients:

• Coefficient of \(-4\sqrt{u}\) is \(-4\)
• Coefficient of \(-3\sqrt{u}\) is \(-3\)

The sum \(-4\sqrt{u} - 3\sqrt{u}\) yields \(-7\sqrt{u}\).

The final result of the expression, therefore, is \(u - 7\sqrt{u} + 12\). Simplifying through combining like terms helps distill the equation into its most condensed form, making it cleaner and more straightforward to handle in further algebraic operations.