Problem 68 - Algebra and Trigonometry

Question

Simplify the radical expressions in Exercises \(67-74\) if possible. $$ \sqrt[3]{150} $$

Step-by-Step Solution

Answer

The cube root of 150 cannot be simplified and remains as \( \sqrt[3]{150} \).

Step 1: Prime Factorization of 150

First, let's do a prime factorization of 150. Begin with the smallest prime number, which is 2, and proceed until you get a prime number. 150 can be written as the product of 2, 3, 5, and 5 (2 * 75 = 150, 3 * 25 = 75, 5 * 5 = 25) So, we can write \(150 = 2 × 3 × 5^2\). None of these factors occur more than three times, so the cube root cannot be simplified.

Step 2: Inserting the Prime Factors into the Radical Expression

Now, replace 150 in the original radical expression with the prime factors. So, the cube root of 150 becomes \( \sqrt[3]{2 \times 3 \times 5^2}\). Since none of the prime factors occurs three times, the radical remains as it is.

Step 3: Conclusion

After attempting to simplify the radical, it is observed that the original radical \(\sqrt[3]{150}\) is in its simplest form because no element appears three times in the expression. Consequently, the cube root of 150 cannot be further simplified.

Key Concepts

Prime Factorization

Understanding how to simplify radical expressions begins with prime factorization, a method to express a number as a product of prime numbers. A prime number is a number greater than 1 that can only be divided by 1 and itself without leaving a remainder.

To perform prime factorization, start with the smallest prime number that divides the number and continue dividing by primes until the number is reduced to 1. Take the given problem \(\sqrt[3]{150} \) as an example: the prime factorization of 150 leads to 2, 3, and 5 as its prime factors with 5 appearing twice, hence we write 150 as \( 2 × 3 × 5^2 \).

Prime factorization simplifies the process of identifying whether the radical can be reduced, as we seek to group the factors into the number of times dictated by the index of the radical—in this case, cube roots looking for groups of three.

Cube Root

A cube root, denoted as \( \sqrt[3]{x} \), is a number that, when multiplied by itself three times, gives the original number x. For instance, if you have a number like 8, the cube root is 2, since \( 2 × 2 × 2 = 8 \).

In the context of simplifying radicals, to simplify a cube root like \( \sqrt[3]{150} \), you would look for a perfect cube among the prime factors. A perfect cube is a number that can be expressed as the cube of an integer, like \( 5^3 = 125 \). However, in our example, since there is no factor that appears three times, there is no perfect cube and therefore the cube root can't be simplified by extracting a factor.

Radical Simplification

Radical simplification is the process of reducing a radical expression to its simplest form. When simplifying cube roots, you need to look for any factor that is raised to the power of three or greater, as these can be taken out of the radical.

In the step-by-step solution of \( \sqrt[3]{150} \), this principle is applied by breaking down 150 into its prime factors, and then looking for triples of the same factor. Since none exists in this case, no further simplification can occur. In practical terms, this means that while some radical expressions can be simplified by removing perfect cubes, others, such as \( \sqrt[3]{150} \) in the given exercise, remain in their original form as the simplest radical form with no repeated factors of three.