Problem 519 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[4]{486 u^{7}}+\sqrt[4]{768 u^{3}}\)

Step-by-Step Solution

Answer

\( \sqrt[4]{3} \sqrt[4]{u^3} (3u \sqrt[4]{2} + 4) \)

Step 1: Express numbers with prime factors

Rewrite the numbers inside the radicals with their prime factors.For 486:\(486 = 2 \times 3^5\)For 768:\(768 = 2^8 \times 3\)So, we have: \( \sqrt[4]{486 u^{7}} + \sqrt[4]{768 u^{3}} = \sqrt[4]{2 \times 3^5 u^{7}} + \sqrt[4]{2^8 \times 3 u^{3}} \)

Step 2: Distribute the radical

Apply the property \( \sqrt[4]{a \times b} = \sqrt[4]{a} \times \sqrt[4]{b} \) to separate the factors.\( \sqrt[4]{2 \times 3^5 u^{7}} = \sqrt[4]{2} \times \sqrt[4]{3^5} \times \sqrt[4]{u^{7}} \)\( \sqrt[4]{2^8 \times 3 u^{3}} = \sqrt[4]{2^8} \times \sqrt[4]{3} \times \sqrt[4]{u^{3}} \)

Step 3: Simplify the radicals

Simplify each part separately.\( \sqrt[4]{2} \) remains as \( \sqrt[4]{2} \).\( \sqrt[4]{2^8} = 2^2 = 4 \).\( \sqrt[4]{3^5} = 3 \times \sqrt[4]{3} \).For the variables:\( \sqrt[4]{u^7} = u^1 \times \sqrt[4]{u^3} = u \sqrt[4]{u^3} \).\( \sqrt[4]{u^3} \) remains as \( \sqrt[4]{u^3} \).

Step 4: Combine the simplified terms

Replace back the simplified terms in the expression:\( \sqrt[4]{2 \times 3^5 u^{7}} = \sqrt[4]{2} \times 3 \sqrt[4]{3} \times u \sqrt[4]{u^3} \)\( = 3u \sqrt[4]{2} \sqrt[4]{3} \sqrt[4]{u^3} \)\( \sqrt[4]{2^8 \times 3 u^{3}} = 4 \sqrt[4]{3} \sqrt[4]{u^3} \)Thus, the expression becomes:\( 3u \sqrt[4]{2} \sqrt[4]{3} \sqrt[4]{u^3} + 4 \sqrt[4]{3} \sqrt[4]{u^3} \)

Step 5: Factor common terms

Factor out the common terms \( \sqrt[4]{3} \sqrt[4]{u^3} \):\( = \sqrt[4]{3} \sqrt[4]{u^3} (3u \sqrt[4]{2} + 4) \)

Key Concepts

prime factorization

Prime factorization is a method used to express a number as the product of its prime factors. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. To break down a number, start by dividing it by the smallest prime number (2) and continue the division process until all factors are prime. In our exercise, the numbers 486 and 768 are broken down into prime factors for ease of calculation. For example:

- 486 can be broken down into: \(2 \times 3^5\).
- Similarly, 768 breaks down into: \(2^8 \times 3\).
Understanding prime factorization is essential as it simplifies complex expressions and helps in further mathematical operations.

fourth roots

The fourth root of a number is a value that, when raised to the power of 4, gives the original number. In mathematical notation, the fourth root of x is represented as \(\sqrt[4]{x}\) or \(x^{1/4}\). This exercise involves simplifying expressions that contain fourth roots.

When simplifying, use the property \(\sqrt[4]{a \times b} = \sqrt[4]{a} \times \sqrt[4]{b}\) to separate the terms inside the radical.

Here's a breakdown:
- \(\sqrt[4]{486u^7}\) becomes \(\sqrt[4]{2} \times \sqrt[4]{3^5} \times \sqrt[4]{u^7}\).
- \(\sqrt[4]{768u^3}\) becomes \(\sqrt[4]{2^8} \times \sqrt[4]{3} \times \sqrt[4]{u^3}\).
By breaking the expression down, each term can be simplified individually and reassembled to form the final expression.

simplification

Simplification involves reducing an expression to its simplest form, making it easier to understand or solve. In the exercise, we simplify the radicals of both numbers and variable parts step-by-step.

To simplify the radicals, apply the following processes:

- Separate the factors using distributive property: \(\sqrt[4]{a \times b}\) to \(\sqrt[4]{a} \times \sqrt[4]{b}\).
- For \(\sqrt[4]{2^8}\), simplify to \(2^2 = 4\), because \(2^8 = (2^4)^2\).
- For the variables, use properties of exponents. For example, \(\sqrt[4]{u^7}\) becomes \(u^{7/4}\) and can be simplified further for easier calculation.

Once all parts are simplified, reassemble the terms to form the simplified expression. The final factorization step—combining common terms—creates a more manageable and understandable expression.

algebraic expressions

An algebraic expression is a mathematical phrase that can include numbers, variables (like u), and operations (like addition and multiplication). Simplifying algebraic expressions involves combining like terms and simplifying radicals or exponents, as seen in our exercise.

Here's a breakdown:

- Combine like terms after simplifying the radicals. In the exercise, after separating and simplifying, each term is recombined.
- For instance, once the radicals are simplified—and any like terms factored out—the expression \(3u \times \sqrt[4]{2} \times \sqrt[4]{3} \times \sqrt[4]{u^3} + 4 \times \sqrt[4]{3} \times \sqrt[4]{u^3}\) is factored to create \(\sqrt[4]{3} \times \sqrt[4]{u^3} (3u \sqrt[4]{2} + 4)\).

This makes the expression much simpler to work with and easier to understand.