Problem 68 - College Algebra

Question

Simplify the radical expressions if possible. $$\sqrt[3]{150}$$

Step-by-Step Solution

Answer

\(\sqrt[3]{150}\)

Step 1: Prime Factorization of 150

Express 150 as a product of its prime factors. It can be done by dividing 150 by prime numbers (2, 3, 5, 7, 11, ...) starting from 2 up to the point where no further division is possible. Doing this provides the prime factors of 150 as \(2 \times 3 \times 5^2\).

Step 2: Grouping the Factors

According to the properties of radicals, the cubic root can simplify to an expression \(x\) if we can group the factors into three similar terms under the root sign. Here in \(2 \times 3 \times 5^2\), we can't find any such group of three similar factors, so the expression can't be simplified further.

Step 3: Write down the Simplified Form

Since there are no groups of three similar factors, the simplified form remains as the original, which is \(\sqrt[3]{150}\).