Problem 82 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

18

Step 1: Finding the square root of 4

The expression \( \sqrt{4} \) can be simplified by finding a number that when multiplied by itself equals 4. This number is 2, so \( \sqrt{4} \) becomes 2.

Step 2: Finding the square root of 9

Similarly, you can simplify \( \sqrt{9} \) by finding a number that when squared equals 9. This number is 3, so \( \sqrt{9} \) becomes 3.

Step 3: Multiplying the entire expression

Now, return to the original expression and substitute \( \sqrt{4} \) with 2 and \( \sqrt{9} \) with 3. The original expression \( \sqrt{4} \cdot 3 \cdot \sqrt{9} \) is then written as \( 2 \cdot 3 \cdot 3 \) which when multiplied equals 18.

Key Concepts

Square Root Calculation

Understanding how to calculate square roots is fundamental to simplifying algebraic expressions involving radicals. A square root, represented by the symbol \( \sqrt{} \), is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{4} \) asks us to find the number which, when squared (multiplied by itself), equals 4.
The process of finding this number is known as 'square rooting'. To simplify \( \sqrt{4} \), we search for a number which gives 4 when squared. The answer, in this case, is 2, because \( 2 \times 2 = 4 \). This process is repeated for any square root calculation, such as \( \sqrt{9} \), where the answer is 3, because \( 3 \times 3 = 9 \).
When dealing with larger numbers or numbers that are not perfect squares, you can break them down into prime factors or use a calculator. For students, remembering the square roots of numbers up to at least 12 can help speed up this process and is a handy tip for simplifying expressions efficiently.

Multiplying Radicals

Once individual square roots are simplified, we often need to perform operations with them, such as multiplication. Multiplying radicals, which are expressions that contain a root, follows the same basic principles as multiplying regular numbers. However, it's important to remember that you can only multiply radicals with the same index, which is the little number written in the crook of the square root symbol. For the majority of cases in basic algebra, we deal with the square root, which has an implied index of 2.
In our exercise example, we have the simplified forms of \( \sqrt{4} \) and \( \sqrt{9} \), which are 2 and 3, respectively. To multiply these radicals, now that they are simplified to whole numbers, you multiply them like any other whole numbers: \( 2 \times 3 \times 3 = 18 \).
Remember that before multiplying, always simplify the radicals to their most reduced form if possible. This practice ensures your final multiplication is as straightforward as a basic arithmetic operation, and it's a key step in multiplying radicals effectively.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols, such as plus, minus, multiplication, and division. Simplifying these expressions is a frequent task in algebra.
To simplify the expression \( \sqrt{4} \cdot 3 \sqrt{9} \), the process involves both the calculation of square roots and the multiplication of radicals, as shown in our step-by-step solution. This expression simplification gives a clear example of how algebra often requires the combination of several skills. Always begin simplification by resolving the operations within the radicals before addressing other numerical operations or variables present in the expression.
It is through the proper understanding of each component within an algebraic expression that one can conduct operations with confidence. Emphasizing clarity in each step, such as identifying and simplifying square roots and handling radical multiplication appropriately, will streamline the process and lead to a correct and simplified result.