Problem 20 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(\sqrt[3]{216}\)

Step-by-Step Solution

Answer

The cube root of 216 is 6.

Step 1: Understand the Expression

The expression \(\sqrt[3]{216}\) represents the cube root of 216. This means we need to find a number which, when multiplied by itself three times, equals 216.

Step 2: Factorization of 216

To find the cube root, first factor the number 216 into its prime factors.\[216 = 2 \times 108 = 2 \times 2 \times 54 = 2 \times 2 \times 2 \times 27 = 2^3 \times 3^3\]Thus, 216 can be expressed as \(2^3 \times 3^3\).

Step 3: Extract the Cube Root

Since 216 is \(2^3 \times 3^3\), the cube root \(\sqrt[3]{216}\) is the product of the cube roots of \(2^3\) and \(3^3\). This results in: \(\sqrt[3]{2^3} \times \sqrt[3]{3^3} = 2 \times 3 = 6\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are those greater than 1 and have no divisors other than 1 and themselves. For the number 216, we find its prime factors by dividing by the smallest prime number possible and continuing the process until we're left with only prime numbers.
- Start with 216, and you see it's even, meaning it has 2 as a factor. - Divide by 2 to get 108. Again, 108 is even, so divide by 2 to get 54. - Repeating this gives you 27, which is not even, so you switch to the next smallest prime number, 3. Divide 27 by 3 three times to get 1. - This results in the prime factorization of 216 as \[216 = 2^3 \times 3^3\] Understanding prime factorization is essential as it makes it easier to simplify expressions involving roots and powers.

Extracting Roots

Extracting roots, in simple terms, is the reverse operation of raising a number to a power. Specifically, when you're dealing with cube roots (\( \sqrt[3]{} \)), you're looking for a number that when multiplied by itself three times gives the original number. With prime factorization, extracting cube roots becomes simpler.
- Take the prime factors found previously, which were \(2^3 \times 3^3\). - Recognize that a cube root over a cube power results in the base itself: \( \sqrt[3]{2^3} = 2\) and \( \sqrt[3]{3^3} = 3\). - Therefore, \( \sqrt[3]{216} = 2 \times 3 = 6\). By understanding root extraction through prime factorization, you can handle larger numbers and complex expressions with ease.

Powers and Roots

Powers and roots are two sides of the same mathematical concept. Powers, like squaring or cubing a number, involve multiplying a number by itself a certain number of times. Roots, such as square roots or cube roots, undo this process, bringing a number back to its original base.
- To handle powers and roots effectively, it is important to become comfortable with exponent rules. - For instance, \(a^{m/n}\) can be seen as the \(n^{th}\) root of \(a^m\), highlighting the relationship between power and roots. - In the exercise, taking the cube root of \(2^3 \times 3^3\) led to simply multiplying the bases: \(2 \times 3\). By mastering the interplay between these operations, you can develop a deeper understanding of algebraic concepts and arithmetic operations.