Question

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[4]{32 x}+\sqrt[4]{2 x^{5}}$$

Short answer

\root{4}{2x} (2 + x)

Step-by-step solution

Break down each term under the fourth root

First, identify the individual components under each fourth root. The given expression is \(\root{4}{32x} + \root{4}{2x^5}\). Write each term in its prime factorized form. \[ 32 = 2^5 \text{ so } \root{4}{32x} = \root{4}{2^5 x} \text{ and } 2x^5 \text{ stays the same.} \]

Simplify the fourth roots

Evaluate the fourth roots by factoring out terms with powers of four: \(\root{4}{2^5 x} + \root{4}{2 x^5}\). \[ 2^5 = 2^4 \times 2^1 \text{ and } x = x \text{ so } \root{4}{2^5 x} = 2 \root{4}{2x}\] and \[2 x^5 = 2 x^4 \times x \text{ so } \root{4}{2 x^5} = x \root{4}{2x}. \]

Combine the simplified terms

The simplified terms from previous steps are \[2 \root{4}{2x} + x \root{4}{2x}. \] Notice that both terms share the common factor \(\root{4}{2x}\).

Factor out the common term

Factor out \(\root{4}{2x}\) from both terms: \[2 \root{4}{2x} + x \root{4}{2x} = \root{4}{2x} (2 + x). \]

Key concepts

Fourth Roots

Understanding the concept of fourth roots is crucial for simplifying expressions like \(\root{4}{32x} + \root{4}{2x^5}\). A fourth root is essentially the number that, when multiplied by itself four times, gives the original number. For instance, the fourth root of 16 is 2, because \(2 \times 2 \times 2 \times 2 = 16\).

When simplifying expressions involving fourth roots, it helps to break down numbers into their prime factors. This way, you can identify which parts can be simplified further. Let's look at the term \(\root{4}{32}\). Since \({32 = 2^5}\), its prime factorized form is easier to handle.

Prime Factorization

Prime factorization involves breaking down a number into its prime numbers, the fundamental building blocks of all integers. For example, 32 can be factorized into \(2^5\).

In the original exercise, we prime factorize 32 to make the expression inside the fourth root more manageable. Here's how it looks: we rewrite \(32x\) as \(2^5 \times x\). Similarly, for the second term in the expression, \(2x^5\), we already have the prime factors exposed.

Factoring Out Common Terms

Factoring out common terms simplifies expressions significantly. In our given problem, after simplifying the fourth roots, we ended up with \(2 \root{4}{2x} + x \root{4}{2x}\).

Both terms contain the common factor \( \root{4}{2x} \). We factor this out to get a more simplified and consolidated expression: \(\root{4}{2x}(2 + x)\).

This step is crucial because it makes the final expression cleaner and easier to understand.

Evaluation of Roots

Evaluating roots, especially fourth roots, requires breaking down the expression into simpler parts. For instance, in \(\root{4}{2^5 x}\), we express \(2^5\) as \(2^4 \times 2 \). The fourth root of \(2^4\) is 2, so the expression simplifies to \(\root{4}{2^5 x} = 2 \root{4}{2x}\).

Similarly, in \(\root{4}{2x^5}\), we recognize \(2x^5 = 2x^4 \times x \). The fourth root of \(x^4\) is x, leading to simplification: \(\root{4}{2x^5} = x \root{4}{2x}\).

Evaluating roots in this manner breaks down complex expressions into more understandable chunks, making the entire simplification process much easier.