Problem 13 - Algebra 2

Question

Simplify. Assume that all variables are positive. $$ \sqrt[3]{54 y^{10}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{54 y^{10}} \) is \(3y^{3} \sqrt[3]{2y^{1/3}} \).

Step 1: Prime Factorize the Numeric Part

Find the prime factorization of 54 which is \(2 \times 3^{3}\). So the cubed root of 54 is 3, because \(3^{3} = 27\), which is close to 54.

Step 2: Simplify the Cubed Root of the Variable Part

The cubed root of \( y^{10} \) can be simplified as \( y^{10/3} = y^{3} \times y^{1/3} \). The first part, \( y^{3} \), can come out of the cubed root because it is perfectly under cubed root. The second part, \( y^{1/3} \), stays under the cubed root because it is not perfectly under cubed root.

Step 3: Write the Final Simplified Form

The final simplified form of the cubed root would combine the results of Step 1 and Step 2. This would be \(3y^{3} \sqrt[3]{2y^{1/3}} \).

Key Concepts

Cube Roots

A cube root is a number that, when multiplied by itself three times, produces the original number. For example, the cube root of 27 is 3, since \(3 \times 3 \times 3 = 27\).
Cube roots are represented using the symbol \(\sqrt[3]{\cdot}\). To calculate the cube root of any number, you need to understand the principle of reversing cube operations.

• If you know the prime factorization of a number, extracting cube roots becomes easier.
• Remember, cube roots deal with cubes, not squares like in square roots.
• You extract groups of three when simplifying cube roots.

Let’s see how this applies in our example with \( \sqrt[3]{54} \). Since the prime factorization of 54 is \(2 \times 3^3\), the cube root of 54 can simplify by taking one \(3\) out, because \(3\times 3\times 3 = 27\) and 27 is about half of 54. You also have a remainder that stays inside the root, like \(\sqrt[3]{2}\).

Simplifying Radicals

Simplifying radicals involves breaking down the expression under the radical symbol into its simplest form. The simpler the form, the easier it is to work with.
When simplifying cube roots and other radicals, aim to make the expression as neat as possible, primarily through factorization.

• Find perfect cubes or squares hidden inside the expression.
• Remove these perfect values from under the radical sign.
• What can't be simplified stays under the radical.

For example, let’s simplify \( \sqrt[3]{y^{10}} \). The cube root translates to dividing the exponent by three. Simplifying gives \(y^{3} \times y^{1/3} \). The \(y^{3}\) can be extracted completely from the radical because it's a perfect cube. But \(y^{1/3}\) remains under the root as it doesn't divide evenly by 3.

Prime Factorization

Prime factorization is crucial when dealing with radical expressions. It involves breaking a number into its smallest prime factors. For example, identifying that 54 breaks down into \(2 \times 3^3\). Knowing these factors lets us simplify cube roots or other roots effectively.
Prime factorization is about expressing a number or variable as a product of prime numbers.

• Start dividing the number by the smallest prime (like 2), then move to the next primes if needed (like 3, 5, etc.).
• This process continues until the number is expressed entirely as a product of primes.
• For variables, the same principle of breaking them down by exponents applies.

Remember, prime factorization helps in easily identifying perfect cubes, which simplifies the extraction process when dealing with cube roots.