Problem 54 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Factor the expression inside the square root

Begin by factoring both the number and the variable part inside the square root. To factor 147, find its prime factors: 147 can be broken down into 3, 49, and 49 can further be broken down into 7 and 7. So, 147 = 3 × 7². For the variable part, we have \(k^3 = k^2 \times k\). Therefore, the expression under the square root becomes \(3 \times 7^2 \times k^2 \times k\).

Step 2: Apply the square root to each factor

Use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Apply this property to each factor: \(\sqrt{3 \times 7^2 \times k^2 \times k} = \sqrt{3} \times \sqrt{7^2} \times \sqrt{k^2} \times \sqrt{k}\).

Step 3: Simplify the square roots of perfect squares

Evaluate the square roots of the perfect squares: \(\sqrt{7^2} = 7\) and \(\sqrt{k^2} = k\). Substitute back into the expression: \(7 \times k \times \sqrt{3} \times \sqrt{k}\).

Step 4: Combine remaining square roots and simplify

Combine the remaining square roots: \(\sqrt{3} \times \sqrt{k} = \sqrt{3k}\). The expression is now \(7k \times \sqrt{3k}\). So, the simplified form is \(7k\sqrt{3k}\).

Key Concepts

Prime Factorization

Prime factorization is the process of finding which prime numbers multiply together to make the original number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In the context of simplifying radicals, prime factorization is essential.
Understanding prime factorization can make the simplification process straightforward. Here's a quick way to start factoring any number:

• Begin with the smallest prime, which is 2. If the number is even, it's divisible by 2.
• Continue dividing by 2 until you can't anymore, then move to the next smallest prime, which is 3. Repeat the process.
• Continue this method using successive prime numbers (5, 7, 11, etc.) until you have only prime numbers left.

Let's apply this to the number 147 from our example. We divide 147 by 3 to get 49, and then break 49 down into 7 and 7. Hence, the prime factorization of 147 is 3 and 7². This breakdown simplifies the calculation of roots, as seen in later steps.

Square Roots

The square root of a number represents a value that, when multiplied by itself, gives the original number. For instance, \(\sqrt{9} = 3\) because 3 times 3 is 9. Similarly, the process is applied when simplifying square roots of variables.
Here are some basics about square roots you should know:

• A perfect square is a number that has a whole number as its square root.
• The square root symbol is represented as \(\sqrt{}\).

In our exercise, we used the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to manage the square root of composite numbers and variables. This method is especially useful when each component inside the square root has factors that are perfect squares. For example, \(\sqrt{49} = 7\) since 49 is a perfect square. This simplification helps in breaking down complex expressions into more manageable parts.

Expressions

An expression in mathematics consists of numbers, variables, and operations combined to represent a particular value. Simplifying an expression means reducing it to its most straightforward form without changing its value.
When dealing with expressions inside square roots, here's a quick guide:

• Identify and factor any numbers into their prime constituents.
• Recognize which parts of the expression can be simplified, especially if you identify perfect squares.
• Use the property that allows breaking the square roots into individual parts, simplifying each component separately where possible.

For instance, in our exercise \(\sqrt{147k^3}\), the expression is broken into prime factors which simplify parts inside the root that contain perfect squares. By isolating these segments, simplification becomes more manageable, allowing the numbers to be expressed outside the square root when possible. This technique ensures that the expression remains efficient and straightforward.

Perfect Squares

Perfect squares are numbers that result from an integer multiplied by itself. Understanding perfect squares is crucial in simplifying radicals because they allow parts of an expression to be taken out of the square root more easily.
Important points about perfect squares:

• Numbers like 1, 4, 9, 16, 25, and so forth are perfect squares because they are squares of whole numbers.
• Recognizing these numbers in the prime factorization process enables quick simplification.
• Perfect squares simplify into their root value, reducing complexity.

In the context of our exercise, \(7^2\) is a perfect square, allowing \(\sqrt{49} = 7\). Similarly, \(k^2\) is a perfect square, simplifying to \(k\). Identifying and using perfect squares is a powerful step towards simplifying radicals like in \(7k\sqrt{3k}\), achieving a cleaner and more concise expression.