Problem 52 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{54 x^{3}}\)

Step-by-Step Solution

Answer

Simplified radical form is \(3x\sqrt[3]{2}\).

Step 1: Factor under the radical

\(54x^3 = 27 \cdot 2 \cdot x^3 = 3^3 \cdot 2 \cdot x^3\)

Step 2: Simplify

\(\sqrt[3]{54x^3} = \sqrt[3]{3^3 \cdot x^3 \cdot 2} = 3x\sqrt[3]{2}\)

Key Concepts

Cube Roots

Cube roots are similar to square roots, but instead of finding a number that, when multiplied by itself twice, gives the original number, you are looking for a number that, when multiplied by itself three times, results in the original number. For example, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\). To simplify expressions with cube roots, you can factor the number under the root to identify any perfect cubes. In the problem \( \sqrt[3]{54x^3} \), notice that we're looking for a cube root. It's essential to break the number into a product of its smallest parts (prime factors) to simplify this expression effectively.

Prime Factorization

Prime factorization is the process of expressing a number as a product of prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To factor the number 54, you would repeatedly divide by prime numbers starting from the smallest, which is 2. Here is how you do it:

• Divide 54 by 2 (the smallest prime number) to get 27.
• Since 27 is not divisible by 2, move to the next prime number, which is 3.
• Divide 27 by 3 to get 9.
• Then divide 9 by 3 to get 3.
• You end up with 3 as a prime number.

This gives us the prime factorization of 54: \(2 \times 3^3\). Recognizing these factors makes it straightforward to identify and simplify terms in radical expressions.

Expressions with Variables

When simplifying expressions that involve variables, remember they can often be treated just like numerical terms. In \(\sqrt[3]{54x^{3}}\), you have both a numerical factor and a variable part. The variable \(x^3\) is a perfect cube, which simplifies nicely when taking a cube root.Here's how variables are handled in such expressions:

• Since the cube root of \(x^3\) is simply \(x\) (because \(x \times x \times x = x^3\)), \(x^3\) can be taken out from under the radical sign as \(x\).

Together with the prime factorization from the numerical part, combining these gives you the simplified radical form. Once you've simplified both parts, you multiply them back together to find the simplest form: \(3x\sqrt[3]{2}\). Understanding how to manage variables in radical expressions helps simplify and solve complex algebraic expressions.