Problem 59 - Elementary Algebra

Question

\(\sqrt[3]{216}\)

Step-by-Step Solution

Answer

The cube root of 216 is 6.

Step 1: Understanding the Problem

Identify the problem, which is to find the cube root of 216, written as \(\textstyle \sqrt[3]{216}\textstyle\). The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Step 2: Factorize the Number

Factor 216 into its prime factors. Start by dividing by the smallest prime numbers: \(\textstyle 216 \div 2 = 108 \). Continue: \(\textstyle 108 \div 2 = 54 \), \(\textstyle 54 \div 2 = 27 \), \(\textstyle 27 \div 3 = 9 \), \(\textstyle 9 \div 3 = 3 \), \(\textstyle 3 \div 3 = 1 \). Therefore, the prime factorization of 216 is \(\textstyle 2^3 \times 3^3 \).

Step 3: Apply the Cube Root

Use the properties of exponents to simplify: \(\textstyle \sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3}\textstyle\). Since both 2 and 3 are raised to the power of 3, the cube root can be distributed: \(\textstyle \sqrt[3]{2^3} \times \sqrt[3]{3^3} = 2 \times 3 = 6\textstyle\).

Step 4: Final Answer

Combine the results from the previous step to find the cube root: \(\textstyle \sqrt[3]{216} = 6\textstyle\).

Key Concepts

Prime Factorization

Prime factorization is breaking down a number into its smallest prime numbers. Primes are numbers greater than 1 that have no divisors other than 1 and themselves. For example, 2, 3, 5, 7, and 11 are prime numbers. To factorize 216:

• Divide by the smallest prime, 2: \(216 \div 2 = 108\).
• Continue dividing by 2: \(108 \div 2 = 54\).
• Then, \(54 \div 2 = 27\).
• Switch to the next prime, 3: \(27 \div 3 = 9\).
• Next, \(9 \div 3 = 3\).
• Finally, \(3 \div 3 = 1\).

So, \(216 = 2^3 \times 3^3\).This breaks 216 into prime factors.

Properties of Exponents

The properties of exponents are rules that apply when working with exponential expressions. Important rules include:

• Power of a Power: \((a^m)^n = a^{mn}\).
• Product of Powers: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: \(a^m \div a^n = a^{m-n}\).
• Cube Root of Powers: The cube root of a power simplifies the exponent: \(\sqrt[3]{a^3} = a\).

For example, \(\sqrt[3]{2^3 \times 3^3} = \sqrt[3]{2^3} \times \sqrt[3]{3^3} = 2 \times 3 = 6\). This makes it easy to simplify the cube root of products involving exponents.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the radical sign to its simplest form. Radicals include square roots, cube roots, and more.

• Identify Perfect Powers: Find numbers that are perfect squares or cubes. In the case of cube roots, perfect cubes are key.
• Prime Factorization: Break the number into prime factors. This helps in identifying perfect cubes within the number.
• Apply the Root: Use properties of exponents to simplify. For example, \(\sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3} = \sqrt[3]{2^3} \times \sqrt[3]{3^3} = 2 \times 3 = 6\).

The cube root of 216 simplifies to 6 because 216 is a perfect cube (\(216 = 6^3\)). Look for such patterns to make simplifying radicals easier.