Question

Simplify the expression. $$\sqrt{4} \cdot 3 \sqrt{9}$$

Short answer

The simplified version of the expression \(\sqrt{4} \cdot 3 \sqrt{9}\) is 18.

Step-by-step solution

Simplify the Square Roots

In the given expression \(\sqrt{4} \cdot 3 \sqrt{9}\), both \( \sqrt{4} \) and \(\sqrt{9} \) can be simplified as these are perfect squares. So, simplifying these gives 2 and 3 respectively.

Perform the Multiplication

After simplification, the expression becomes \(2 \cdot 3 \cdot 3\). Now, the multiplication operation is performed over these numbers which results in an integer.

Key concepts

Understanding Perfect Squares

When we talk about perfect squares, we are referring to the products of an integer multiplied by itself. For example, 4 is a perfect square because it can be derived from 2 times 2. Simplifying square roots, therefore, becomes quite straightforward when dealing with perfect squares. In our original exercise, both 4 and 9 are perfect squares as they are the squares of 2 and 3 respectively.

To simplify square roots of perfect squares, one must identify the number that, when multiplied by itself, results in the radicand (the number under the square root symbol). In our case, \( \sqrt{4} \) simplifies to 2 and \( \sqrt{9} \) simplifies to 3 because \( 2 \times 2 = 4 \) and \( 3 \times 3 = 9 \). Recognizing these perfect squares and their roots can help streamline the process of simplifying square roots in algebraic expressions.

Simplifying Square Root Operations

The square root operations pertain to the procedures involved in manipulating square roots within equations or expressions. Simplifying a square root means finding its most basic, non-radical form, if possible. The first step in dealing with square roots is as illustrated in our original exercise: look for perfect squares and simplify them.

Following the simplification, it's crucial to apply the fundamental operations such as multiplication, as in \(2 \cdot 3 \cdot 3\) from our example. Remember, after simplifying the square roots, what remains should be treated like any other numerical or algebraic term. Operations involving addition, subtraction, multiplication, and division follow the usual arithmetic rules. Using these guidelines helps students methodically approach square root problems and reduces the likelihood of error in their computation.

Multiplication of Radicals

The multiplication of radicals involves a couple of rules that make the operation straightforward. When multiplying radicals, if the indices (the small number to the left of the radicand, which tells you the degree of the root) and the radicand (the number under the radical sign) are the same, you can multiply the radicands together under one single radical. However, our exercise deals with a simpler case, where we multiply a radical by an integer.

Once you have the square roots simplified to integers, as with \( \sqrt{4} \) and \( \sqrt{9} \) to 2 and 3 respectively, you multiply them like any other integers. In this case, we multiply 2 by the product of 3 and 3, resulting from the simplified radical 3 \( \sqrt{9} \). The key here is to simplify the radicals before multiplying, which allows for more accessible and accurate computations and contributes to a student's growing fluency in algebraic operations involving radicals.