Problem 54 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{147 k^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(7k\sqrt{3k}\).

Step 1: Factor the Radicand

Begin by factoring the expression inside the square root, which is \( 147k^3 \).\[ 147 = 3 \times 49 = 3 \times 7^2 \]\[ k^3 = k^2 \times k \]This gives us:\[ \sqrt{147k^3} = \sqrt{3 \times 7^2 \times k^2 \times k} \]

Step 2: Simplify the Radicand

Apply the property of square roots that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \):\[ \sqrt{3 \times 7^2 \times k^2 \times k} = \sqrt{3} \times \sqrt{7^2} \times \sqrt{k^2} \times \sqrt{k} \]We can simplify further by taking the square roots of perfect squares:\[ = \sqrt{3} \times 7 \times k \times \sqrt{k} \]

Step 3: Combine Simplified Terms

Combine the simplified terms into one expression:\[ \sqrt{147k^3} = 7k\sqrt{3k} \]

Key Concepts

Factoring Radicands

Factoring radicands is a crucial technique to simplify expressions under a square root. When you factor a radicand, you break it down into its prime factors or numbers that multiply together to give the radicand. For instance, to factorize \( 147 \), recognize that \( 147 = 3 \times 49 \) and \( 49 = 7^2 \), hence \( 147 = 3 \times 7^2 \). Similarly, if you have a variable, like \( k^3 \), think of it as \( k^2 \times k \).

• Decompose the number into prime factors.
• Compare all factors of the expression to identify potential perfect squares.
• Remember to handle both numeric and variable parts separately.

Breaking the expression down in this manner helps in applying the next steps to simplify it further.

Properties of Square Roots

Understanding the properties of square roots is essential for simplifying expressions effectively. One of the key properties is that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This property is particularly useful when dealing with square roots of products, as it allows the roots to be pulled apart.The property comes in handy when splitting the radicand into recognized perfect squares:

• If \( 49 = 7^2 \), then \( \sqrt{49} = 7 \).
• If \( k^2 \) is inside the square root, \( \sqrt{k^2} = k \).

This means, for example, that \( \sqrt{3 \times 7^2 \times k^2 \times k} \) becomes \( \sqrt{3} \times 7 \times k \times \sqrt{k} \). Recognizing how to separate and simplify using this property can significantly simplify your calculations.

Perfect Squares

Perfect squares are numbers or expressions that can be written as the square of an integer or another expression. Recognizing perfect squares within radicands can help simplify square roots. When you identify a perfect square, you can take its square root easily.

• Examples of perfect squares include \( 4, 9, 16, \text{ and } 25 \) because they are \( 2^2, 3^2, 4^2, \text{ and } 5^2 \) respectively.
• Within variables, \( k^2 \) is considered a perfect square because it is the square of \( k \).

By identifying \( 7^2 \) and \( k^2 \) as perfect squares within the expression \( \sqrt{147k^3} \), you pull them outside the square root easily: \( 7 \times k \). This leads to a simplified expression of \( 7k\sqrt{3k} \). Recognizing and utilizing perfect squares from the start simplifies your work significantly.