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Simplifying square roots, or 'radicals', is a fundamental skill in algebra. To simplify a square root, you aim to find the largest square factor of the number inside the root. For instance, the square root of 27, expressed as \( \sqrt{27} \), can be broken down into \( \sqrt{9 \times 3} \). Since 9 is a perfect square (3 squared), we can take the square root of 9 out, which gives us \( 3 \sqrt{3} \). Similarly, \( \sqrt{147} \) simplifies to \( 7\sqrt{3} \) by recognizing that 147 is divisible by 49, which is 7 squared. Thus, the principle revolves around factoring the radicand (the number under the radical sign) into a product of a square number and another number and then taking the square root of the square number.

This process allows us to work with simplified versions of the radicals, which is essential for further operations like addition, subtraction, or solving equations. Mastering this skill ensures students can efficiently tackle more complex algebraic expressions involving square roots.

When it comes to algebraic expressions, 'combining like terms' is a necessary technique for simplification. Like terms are terms that have the same variable raised to the same power. In the context of square roots, like terms would therefore have the same radicand. After simplifying the radicals in our example, we obtain \( 3\sqrt{3} \) and \( -7\sqrt{3} \). These two are like terms since they both contain the \( \sqrt{3} \) component.

To combine them, we keep the radical part constant and combine the coefficients (the numbers in front of the radical). This is similar to how we would combine terms like 3x and -7x in basic algebra, resulting in -4x. Applying this to our example with square roots, we get \( 3\sqrt{3} - 7\sqrt{3} = -4\sqrt{3} \). It's crucial to highlight that combining like terms is a way to tidy up expressions and it's often one of the final steps in simplifying algebraic expressions.

Square root subtraction involves finding the difference between two square root terms. However, just as you can only subtract like terms, you can only subtract like radical terms. This means that the radicands—the numbers under the square root symbols—must be the same in order to subtract. After simplifying each radical term as we did in the previous sections, if they share the same radicand, you can then proceed to subtract the coefficients.

In the example, after simplification, we have \( 3\sqrt{3} \) and \( 7\sqrt{3} \) which can be subtracted to get \( -4\sqrt{3} \). It is important to maintain the order of terms, as switching them can lead to a sign error, which will give an incorrect result. Grasping square root subtraction is not only key for handling algebraic manipulations but also for understanding concepts in geometry, where square roots often arise.