Problem 17 - Elementary and Intermediate Algebra

Question

Simplify each expression. $$ \sqrt[3]{32} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{32} \) is 2.

Step 1: Identify the Expression

We need to simplify the expression \( \sqrt[3]{32} \), which means we are looking for a number that, when multiplied by itself three times (cubed), gives the number 32.

Step 2: Identify Perfect Cubes

We need to recall or identify any perfect cubes that might be factors of 32. A perfect cube is a number like 1, 8, 27, etc., where each number is the cube of an integer.

Step 3: Write 32 as a Product of its Prime Factors

Start by breaking down 32 into its prime factors: \( 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 \).

Step 4: Group the Prime Factors into a Cube

Since we want a cube root, we look for three identical factors to form a cube. We can write \( 2^5 \) as \( 2^3 \times 2^2 \).

Step 5: Simplify the Expression

The expression \( \sqrt[3]{32} \) can now be simplified by taking the cube root of \( 2^3 \), which is \( 2 \), and leaving \( 2^2 \) outside as it doesn't form a complete cube. Hence, \( \sqrt[3]{32} = 2 \), since \( \sqrt[3]{2^3} = 2 \).

Key Concepts

Simplification

Simplifying an expression often means reducing it to its most basic form while preserving its original value. When dealing with cube roots, this involves breaking down a number to see if it can be expressed in simpler terms. In our exercise, we are looking for the cube root of 32. This means finding a number that, when multiplied by itself three times, equals 32.
Simplification makes it easier to work with numbers, especially when solving further mathematical problems or equations.

• Start by identifying if the number is a perfect cube.
• If not, factor the number to find a simplified form.
• The goal is to express the number as the product of its simplest factors.

Simplifying cube roots can help you recognize patterns and relationships between numbers, which is a key skill in math.

Prime factorization

Prime factorization is a method used to express 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. For instance, 2, 3, 5, and 7 are all primes.

For the number 32, we need to break it down into these prime factors. Begin by dividing 32 by the smallest prime number, which is 2. Continue dividing until you can no longer divide evenly by 2.

• The process goes: 32 ÷ 2 = 16
• 16 ÷ 2 = 8
• 8 ÷ 2 = 4
• 4 ÷ 2 = 2
• Finally, 2 ÷ 2 = 1

Thus, the prime factorization of 32 is expressed as \( 2^5 \).

This factorization allows us to identify groups of three identical factors necessary for determining the cube root.

Perfect cubes

A perfect cube is a number that can be expressed as three identical factors multiplied together. For example, numbers like 8 (since \(2 \times 2 \times 2 = 8\)) and 27 (\(3 \times 3 \times 3 = 27\)) are perfect cubes.

Understanding perfect cubes is essential when dealing with cube roots. Once a number is expressed through its prime factors, like \(2^5\) for 32, you can easily identify perfect cubes within these factors. In this exercise, we sought groups of three identical prime factors within the prime factorization of 32, i.e., \(2^5 = 2^3 \times 2^2 \).

• \(2^3\) is a perfect cube representing our cube root value of 2.
• The leftover \(2^2\) is not a perfect cube and is thus excluded from contributing to the cube root.

Recognizing perfect cubes simplifies finding the cube root, as it allows you to focus only on those parts of a number that form complete groups, making calculations smoother and more efficient.