Problem 54 - College Algebra

Question

Simplify each expression. $$\sqrt{147 k^{3}}$$

Step-by-Step Solution

Answer

\(7k\sqrt{3k}\)

Step 1: Identify the Radicand

The expression inside the square root, known as the radicand, is 147\(k^3\). This is what we will simplify.

Step 2: Prime Factorization of the Numeric Component

To simplify \(\sqrt{147}\), first perform the prime factorization of 147: \(147 = 3 \times 49 = 3 \times 7^2\). This showcases the perfect square (\(7^2\)) that can be taken out of the square root.

Step 3: Simplify the Numeric Component

Extract the perfect square from under the square root: \(\sqrt{3 \times 7^2} = 7\sqrt{3}\). This gives us the simplified numeric part of the expression.

Step 4: Simplify the Variable Component

Consider the variable under the square root, \(k^3\). This can be rewritten as \(k^2 \times k\). Since \(k^2\) is a perfect square, simplify it as \(k\sqrt{k}\).

Step 5: Combine Simplified Components

Combine the simplified numeric and variable components: \(\sqrt{147k^3} = 7\sqrt{3} \times k\sqrt{k} = 7k\sqrt{3k}\).

Key Concepts

Understanding the Radicand

When we talk about simplifying radical expressions, the first term you need to understand is "radicand." The radicand refers to the number or expression inside the radical symbol (√). This is the part we aim to simplify.
In the given exercise, the radicand is 147\(k^3\). For simplification, various techniques like prime factorization and breaking variables into simpler parts are applied to the radicand.
When you learn to simplify the radicand, you will make handling complex expressions much easier.

Exploring Prime Factorization

Prime factorization is a method of expressing a number as a product of its prime factors. This is crucial for simplifying square roots because identifying perfect squares within the radicand can make the process smoother.
In our example, the prime factorization of 147 involves breaking it down into \(3 \times 49\). Further factorizing 49 gives \(7^2\). By identifying this perfect square, 7, you can simplify the square root expression significantly.
The beauty of prime factorization is that it turns a messy number into manageable chunks, enabling us to extract what we need from the radical.

Grasping Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. Recognizing these within a radicand is key for simplification.
During the simplification process, identifying a perfect square like \(7^2\) from the prime factorization of 147 allows us to take the number 7 outside the square root. This step is essential to make the radicand simpler and the expression cleaner.
If a number inside the square root can be expressed as a perfect square, its root can be easily calculated, reducing the complexity of the expression drastically.

Simplifying Variables Under the Radical

Variable simplification under a radical often involves breaking down the exponent of the variable. For example, in \(k^3\), the aim is to identify parts of the expression that are perfect squares—like \(k^2\).
This can be written as \(k^2 \times k\), separating it into a perfect square \(k^2\) and a single \(k\). You can then simplify \(k^2\) as \(k\), pulling it out of the radical, and leaving \(\sqrt{k}\) inside.
This technique of simplifying variables reduces the complexity of radical expressions and is especially useful when handling expressions containing large exponents or variables.