Problem 30 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[4]{48}-\sqrt[4]{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\sqrt[4]{3}\).

Step 1: Simplify the first term

Start by simplifying the expression inside the fourth root, \( \sqrt[4]{48} \). Begin by expressing 48 as its prime factors: \( 48 = 2^{4} \cdot 3 \). This can be rewritten as \( \sqrt[4]{2^{4} \cdot 3} \). Since \( 2^{4} \) is a perfect fourth power, it can be pulled out of the radical as 2. Therefore, \( \sqrt[4]{48} = 2 \cdot \sqrt[4]{3} \).

Step 2: Substitute back into the original expression

Replace \( \sqrt[4]{48} \) in the original expression \( \sqrt[4]{48} - \sqrt[4]{3} \) with the simplified expression from Step 1: \( 2 \cdot \sqrt[4]{3} - \sqrt[4]{3} \).

Step 3: Combine like terms

Notice that both terms have \( \sqrt[4]{3} \) as a common factor. Therefore, you can combine them: \( (2 - 1) \cdot \sqrt[4]{3} = 1 \cdot \sqrt[4]{3} = \sqrt[4]{3} \).

Key Concepts

Fourth Roots

When we talk about fourth roots, it's essential to understand that we are looking to find a number which when multiplied by itself four times gives the original number.
For example, if we take the fourth root of 16, \[ \sqrt[4]{16} = 2 \]because \[2 \times 2 \times 2 \times 2 = 16.\]
In the provided exercise, we dealt with the expression \( \sqrt[4]{48} \). The aim is to simplify it by identifying any parts of 48 that could become a perfect fourth power.

Prime Factorization

Prime factorization is a crucial step in simplifying radicals, including fourth roots. It involves breaking down a number into its most basic prime components.
For example, with the number 48, it can be expressed as \( 48 = 2^4 \times 3 \).
Each of these numbers is a prime number: 2 and 3, since they cannot be divided further except by 1 and themselves.
Once we have the prime factorization, it is easier to see if any groupings can be factored out, which is what got us to simplify \( \sqrt[4]{48} = 2 \cdot \sqrt[4]{3} \) earlier.

Simplifying Expressions

Simplifying expressions involves combining like terms and reducing them to their simplest forms.
This is particularly helpful in making complex expressions easier to work with.
In the step-by-step solution, after simplifying \( \sqrt[4]{48} \) to \( 2 \cdot \sqrt[4]{3} \), we substitute this back into the original expression resulting in \( 2 \cdot \sqrt[4]{3} - \sqrt[4]{3} \).
The key here is recognizing the common factor, \( \sqrt[4]{3} \), between the terms.
By considering this commonality, the expression can be further simplified by factoring out \( \sqrt[4]{3} \), giving \( (2 - 1) \cdot \sqrt[4]{3} = \sqrt[4]{3} \), making the expression more manageable.