Problem 81 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. \(\sqrt{27}\)

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{27}\) is \(3*\sqrt{3}\).

Step 1: Identify the perfect square

The number 27 can be divided perfectly by the square number 9. So, we rewrite 27 as \(9*3\), giving \(\sqrt{9*3}\).

Step 2: Use the property of roots

Next, using the property of roots that states that the square root of the product of two numbers is the same as the product of the square roots of those numbers, we separate the square roots, becoming \(\sqrt{9}*\sqrt{3}\).

Step 3: Simplify the square root

Simplify the square root of 9, resulting in 3. Keep \(\sqrt{3}\) as it is, because it cannot be simplified further. Thus, the final answer is \(3*\sqrt{3}\).

Key Concepts

Square Roots

When we discuss square roots, we're referring to the operation of finding a number that, when multiplied by itself, gives the original number. For instance, if we take the square root of 25, denoted as \( \sqrt{25} \), we are looking for a number which, when squared (multiplied by itself), equals 25. The answer is 5, because \( 5 \times 5 = 25 \).

Understanding square roots is fundamental when working with radical expressions. It helps in simplifying complex numbers and solving equations. Remember that every positive number has two square roots: one positive and one negative. However, when you see the radical symbol (\( \sqrt{} \)), it typically refers to the positive root, also known as the principal square root.

Perfect Square

A perfect square is an integer that can be expressed as the product of an integer with itself. For example, the number 16 is a perfect square because it can be written as \( 4^2 \) or \( 4 \times 4 \). In the context of the problem \( \sqrt{27} \), we look for perfect square factors of 27 to simplify the radical.

Knowing your perfect squares, like \( 1, 4, 9, 16, 25, ... \), is quite useful. It helps you quickly identify factors that can be taken out of the radical, leading to simplification of the expression. Keep in mind that when dealing with roots, recognizing these perfect squares can make the difference between a simple or complex solution.

Property of Roots

The property of roots is particularly handy in simplifying expressions involving radicals. One of these properties tells us that the square root of a product is equal to the product of the square roots of the individual factors, which can be written as \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).

In the given problem, this property allows us to break down \( \sqrt{27} \) into \( \sqrt{9 \cdot 3} \) and then further into \( \sqrt{9} \cdot \sqrt{3} \). This makes it easier to simplify because we recognize \( \sqrt{9} \) as a perfect square, which simplifies to 3. Applying properties of roots effectively reduces complex expressions to simpler ones and is a cornerstone of working with radical expressions.

Radical Simplification

Radical simplification involves reducing a radical expression to its simplest form. It often includes factoring out perfect squares, utilizing properties of roots, and simplifying any numerical coefficients. For the expression \( \sqrt{27} \), we factor 27 into \( 9 \cdot 3 \) because 9 is a perfect square that we can easily take the root of.

Once we apply the property of roots, splitting our expression into \( \sqrt{9} \cdot \sqrt{3} \), we simplify \( \sqrt{9} \) to 3. The \( \sqrt{3} \) remains unchanged because 3 is not a perfect square. The result is then combined as \( 3 \cdot \sqrt{3} \), which is the simplest form of the original expression. Mastering radical simplification is essential for algebra, higher-level math, and various applications within science and engineering.