Problem 18 - Elementary and Intermediate Algebra

Question

Simplify each expression. $$ \sqrt[3]{40} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{40} = 2\sqrt[3]{5} \)

Step 1: Identify the Radical Expression

The problem asks us to simplify the cube root of 40, which is denoted as \( \sqrt[3]{40} \).

Step 2: Prime Factorization of 40

Perform the prime factorization of 40. We find that 40 can be broken down into its prime factors as follows: \[ 40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5 \]

Step 3: Apply the Property of Cube Roots

Use the property of cube roots that allows us to separate the expression into individual roots: \[ \sqrt[3]{2^3 \times 5} = \sqrt[3]{2^3} \times \sqrt[3]{5} \]

Step 4: Simplify Each Factor

Simplify each cube root. Since \(\sqrt[3]{2^3} = 2\), we can write: \[ \sqrt[3]{2^3} \times \sqrt[3]{5} = 2 \times \sqrt[3]{5} \]

Step 5: Final Expression

Combine the simplified terms to get the final simplified expression: \[ \sqrt[3]{40} = 2\sqrt[3]{5} \]

Key Concepts

Radical Expressions

Radical expressions involve roots, which are the opposite of exponents. Instead of multiplying a number by itself a certain number of times, as with exponents, radicals ask you to find the original number that was multiplied. In this exercise, we are working with a cube root, represented by \(\sqrt[3]{x}\).

This means we’re looking for a number that, when multiplied by itself three times, will give us the original number inside the radical. For \(\sqrt[3]{40}\), our goal is to simplify this expression to make it easier to understand and use in further calculations.

Cube roots are commonly represented using the radical symbol, with a small number 3 above it, indicating it's a cube root rather than a square root.

Prime Factorization

Prime factorization is a key step in simplifying radical expressions, especially when dealing with cube roots. It involves breaking down a number into its prime factors – numbers that are only divisible by 1 and themselves. For the number 40, the prime factorization looks like this:

\[ 40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5 \]

This step is crucial because identifying prime factors allows us to simplify the expression under the radical. Cube roots can then be easily worked with since any group of three identical factors simplifies perfectly outside the radical. For example, \(2^3\) simplifies to 2 when taking the cube root. This method helps to clearly see and utilize perfect cubes within the number.

Simplification Steps

The process of simplifying cube roots involves several steps to achieve a final, clean expression. Here's how you do it:

• Identify and apply the property of cube roots: This involves separating the cube root of a product into the product of cube roots. For \(\sqrt[3]{2^3 \times 5}\), it becomes \(\sqrt[3]{2^3} \times \sqrt[3]{5}\).

• Simplify each component: For perfect cubes like \(2^3\), the cube root is straightforward. \(\sqrt[3]{2^3} = 2\). This means it's taken out of the radical while unperfect cubes stay inside.

• Combine the simplified terms: After simplifying, you combine the simplified components to form the final expression: \(\sqrt[3]{40} = 2\sqrt[3]{5}\).

This method of simplification makes the expression easier to manage and understand, providing a simplified and more useful form for further math operations.