Problem 62 - College Algebra

Question

Simplify the radical expressions in Exercises \(61-68\) $$\sqrt[3]{150}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{150}\) is \(\sqrt[3]{150}\).

Step 1: Factorization

Begin by factoring the number 150. The prime factorization of 150 will yield \(2 \times 3 \times 5^2\).

Step 2: Group the factors

For cube roots, we need to group the factors as cubes. Unfortunately, there is no group of three same numbers, so the number inside the cube root cannot be simplified.

Step 3: Writing the Final Answer

Since we are unable to simplify the expression any further, the final answer would remain identical to the original expression, i.e., \(\sqrt[3]{150}\).

Key Concepts

Cube Roots

The concept of cube roots is important when dealing with radical expressions, particularly those involving the number three, also called the index. A cube root seeks to find a number that, when multiplied by itself three times, results in the original number. For instance, the cube root of 8 is 2, because multiplying 2 by itself three times (or 2 x 2 x 2) equals 8. Unlike square roots, cube roots may be applied to both positive and negative numbers. It's as simple as flipping the sign for negative numbers, thanks to the odd number of multiplications necessary. Understanding cube roots enables you to simplify equations and expressions that contain such radicals, even if fully simplifying isn't possible due to specific prime factors.

Prime Factorization

Prime factorization is a critical step in simplifying radical expressions. It involves breaking down a number into its basic building blocks, known as prime numbers. Prime numbers are numbers that have only two divisors: 1 and themselves. Examples include 2, 3, 5, 7, and 11. Decomposing a number like 150 involves finding these prime numbers that multiply together to give the original number. In our example, 150 can be broken down as:

• Start with the smallest prime, 2: 150 divided by 2 equals 75.
• Next, use 3: 75 divided by 3 equals 25.
• Finally, 25 breaks down into 5 times 5, or 5 squared.

Thus, the prime factorization of 150 is 2 x 3 x 5^2. Prime factorization helps in simplifying radicals by identifying potential groups according to the radical's index.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. They represent numbers under a root symbol or radical sign (\(\sqrt{}\)).The key to working with radical expressions is understanding their index. The index tells you how many identical factors you need to remove the radical. For a cube root, the index is 3, which requires grouping prime factors into threes.While simplifying, you separate the factors that fit this requirement. However, when factors do not meet the criteria, as in the case of our cube root of 150, the radical stays as is. This understanding allows for accurate simplification when possible and recognition of when an expression is in its simplest form.Radical expressions may seem complex at first, but they become easier as you get comfortable grouping and simplifying different factors.