Problem 17 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{32} \) is \( 2 \times \sqrt[3]{4} \).

Step 1: Understand the Problem

We are asked to simplify the cube root \( \sqrt[3]{32} \). This means we need to find a number that, when multiplied by itself three times, gives 32.

Step 2: Prime Factorize the Number

First, we need to express 32 as a product of prime factors. 32 can be broken down as follows:\[32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\]

Step 3: Use the Cube Root Property

The cube root property states that \( \sqrt[3]{a^3} = a \). We apply this to the prime factorization of 32:\[\sqrt[3]{32} = \sqrt[3]{2^5}\]

Step 4: Simplify Using the Cube Root Property

From the factorization \( 2^5 = (2^3) \times 2^2 \), we can extract the cube root of \( 2^3 \) as follows:\[\sqrt[3]{2^5} = \sqrt[3]{(2^3) \times 2^2} = 2 \times \sqrt[3]{4}\]

Step 5: Conclusion

The simplest form of \( \sqrt[3]{32} \) thus becomes \( 2 \times \sqrt[3]{4} \). However, for simplification, the complete simplification includes approximating \( \sqrt[3]{4} \) if necessary.In exact form, \( \sqrt[3]{32} = 2 \times \sqrt[3]{4} \), but if needed as a decimal, approximations may be further applied.

Key Concepts

Prime Factorization

To begin simplifying the cube root of a number, we first need to express the given number as a product of its prime factors. Prime factorization involves breaking down a number until all factors are prime.

For example, let's take the number 32. It can be divided by 2 (a prime number) multiple times, as follows:

• 32 divided by 2 equals 16
• 16 divided by 2 equals 8
• 8 divided by 2 equals 4
• 4 divided by 2 equals 2
• And finally, 2 is a prime number

So, the prime factorization of 32 is expressed as:\[32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\]Breaking the number down into its prime factors is crucial because it sets the stage for applying properties of roots, particularly when dealing with cube roots or square roots, as these roots rely on recognizing perfect powers within numbers.

Cube Root Property

The cube root property is an important tool for simplifying expressions involving cube roots. It is based on the principle that if you have a perfect cube inside the radical sign, you can extract the root easily.

The property can be stated as: \( \sqrt[3]{a^3} = a \). This means that if you have a number expressed as a power of three inside the cube root, you can simplify it directly to its base.

Let's apply this to our exercise:
The expression \( \sqrt[3]{32} \) can be rewritten using its prime factorization, \( \sqrt[3]{2^5} \). Once it's broken down into \( 2^5 = (2^3) \times 2^2 \), you can apply the cube root property to the \( 2^3 \) part, pulling out a 2:\[ \sqrt[3]{2^5} = \sqrt[3]{(2^3) \times 2^2} = 2 \times \sqrt[3]{4} \]The cube root of \( 2^3 \) simplifies directly to 2. Understanding this property allows us to simplify radicals involving perfect cubes efficiently and is a powerful technique to use during problem-solving.

Simplifying Radicals

When dealing with radicals, particularly non-perfect cubes, like \( \sqrt[3]{4} \) in this case, full simplification might not be possible without approximation. Instead, finding the simplest radical form is usually sufficient, especially for exact answers.

The objective is to express the radical as simply as possible. Often, this means using properties of exponents and the prime factors identified earlier. In our situation, we've simplified down to \( 2 \times \sqrt[3]{4} \).

• Start with breaking down numbers into prime factors.
• Apply root properties where possible.
• Simplify further if the remaining number is a perfect power.

For radicals that don't simplify to whole numbers, such as \( \sqrt[3]{4} \), it's often left in its radical form to maintain exactness. Alternatively, decimal approximations can be useful for estimations when precision isn't mandatory in an answer. The ability to simplify radicals, especially cube roots or square roots, is key in mathematics to reduce expressions to their simplest, cleanest forms.