Problem 63 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ -6 \sqrt{4} $$

Step-by-Step Solution

Answer

-12

Step 1: Identify the Radical

In the given expression, \(-6 \sqrt{4}\), the radical is \(\sqrt{4}\).

Step 2: Calculate the Square Root

The square root of 4 is 2. So, \(\sqrt{4}\) simplifies to 2.

Step 3: Multiply the Result by -6

Now replace the \(\sqrt{4}\) in the original expression with 2 and multiply it by -6 to get -12.

Key Concepts

Understanding Square Roots

The concept of a square root revolves around finding a number that, when multiplied by itself, yields the original number under the square root symbol. For instance, when we look at the expression \( -6 \sqrt{4} \) from our exercise, our main task is to identify what number multiplied by itself equals 4.

This is a fundamental aspect of simplifying radical expressions: recognizing perfect squares. Perfect squares are numbers like 1, 4, 9, 16, and so on, that have whole numbers as their square roots. Knowing these by heart speeds up the simplification process. Thus, since the square root of 4 is 2 (\( \sqrt{4} = 2 \)), we have easily found the number that, when squared, gives us back our original number 4.

Navigating Through Radical Notation

Radical notation is used to indicate the root of a number and is represented by the symbol \( \sqrt{} \). The number inside the radical symbol is known as the radicand. In our example, 4 is the radicand. Simplifying an expression in radical notation often involves identifying whether the radicand is a perfect square.

To improve understanding, it's helpful to also know that there are other types of roots, such as cube roots (\( \sqrt[3]{} \) ) and fourth roots (\( \sqrt[4]{} \) ), denoted by an index above the radical sign. However, if no index is shown, it's understood to be a square root. Remember that the square root symbol without an index corresponds to the second root, indicating the need to find a number that can be squared to obtain the radicand.

Executing Arithmetic Operations with Radicals

When simplifying radical expressions, we must often perform arithmetic operations such as addition, subtraction, multiplication, or division. In our textbook exercise, the operation needed after simplifying the radical was multiplication. We calculated the square root of 4, yielding 2, and then we multiplied it by the number outside of the radical, in this case, -6, to get our simplified expression \( -6 \times 2 = -12 \).

It is crucial to follow the correct order of operations, typically remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Here, the expression contains no parentheses or exponents to work out first, so we move directly to multiplication. Always simplify the radical first before applying any other arithmetic operation.