Problem 18 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( 2 \times \sqrt[3]{5} \).

Step 1: Recognize the Radical Expression

The given expression is a cube root, which in mathematical terms is represented as \( \sqrt[3]{40} \). This implies we need to find a number that, when raised to the power of 3, gives 40.

Step 2: Factor the Inside Expression

Factor 40 into its prime factors: \( 40 = 2^3 \times 5 \). We express 40 as a product of its prime factors to simplify the cube root.

Step 3: Simplify the Cube Root

Apply the cube root to each factor: \( \sqrt[3]{2^3 \times 5} = \sqrt[3]{2^3} \times \sqrt[3]{5} \). Here, we separate the cube root of the complete cube from that of the leftover factor 5.

Step 4: Calculate Cube Root of Perfect Cubes

The cube root of \( 2^3 \) is 2 because \( 2 \times 2 \times 2 = 8 \): \( \sqrt[3]{2^3} = 2 \).

Step 5: Write the Simplified Expression

Insert the value obtained from Step 4 back: \( 2 \times \sqrt[3]{5} \). This is the simplified form of the original cube root of 40.

Key Concepts

Prime Factorization

Prime factorization involves expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
To factor a number like 40, we start by identifying the smallest prime, which is 2, and check if it divides 40. Since it does, we divide it repeatedly by 2 until it is no longer divisible:

• 40 divided by 2 equals 20
• 20 divided by 2 equals 10
• 10 divided by 2 equals 5

Now, we see that 5 is a prime number, and we can't divide it further by anything other than itself or 1. So, the prime factorization of 40 is expressed as:

• \( 40 = 2^3 \times 5 \)

Using prime factorization helps in simplifying radical expressions by identifying manageable pieces, like perfect cubes in this case, making it easier to apply operations like cube roots.

Radical Expressions

Radical expressions are expressions that contain a root symbol, such as square roots or cube roots. In our case, we're looking at the cube root of 40.
Notation-wise, the cube root is written as \( \sqrt[3]{40} \). This indicates finding a value that, when multiplied by itself three times, results in 40.

• The number under the root is known as the radicand.
• The degree of the root is called the index, which is 3 in cube roots.

Understanding radical expressions is crucial as they frequently appear in mathematical problems, requiring simplification for easier handling. Applying cube roots to each element separately, especially using prime factorization, can significantly simplify the problem.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. In the case of cube roots, our goal is to break down the expression so that only one radical remains, and any perfect cubes are simplified.
Starting with \( \sqrt[3]{40} \) and knowing that \( 40 = 2^3 \times 5 \), we can separate this expression:

• Apply the cube root to each part: \( \sqrt[3]{2^3} \times \sqrt[3]{5} \)

The \( \sqrt[3]{2^3} \) simplifies to 2, since 2 cubed (\( 2 \times 2 \times 2 \)) equals 8, which is a perfect cube.
The leftover part, \( \sqrt[3]{5} \), cannot be simplified further because 5 is not a perfect cube. Therefore, the expression simplifies to:

• \( 2 \times \sqrt[3]{5} \)

Thus, by systematically breaking down the expression using prime factorization, identifying perfect cubes, and applying the radical index, you achieve a cleaner, more understandable form of the radical expression.