Problem 515 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[6]{\frac{128 u^{7}}{v^{3}}}\)

Step-by-Step Solution

Answer

\(2^{7/6} u^{7/6} v^{-1/2}\)

Step 1: Write the expression inside the radical

Start with the given expression in the radical form: \text{\(\sqrt[6]{\frac{128 u^{7}}{v^{3}}}\)}.

Step 2: Factorize the numerator and the denominator

Factorize 128 to its prime factors and write the expression with exponents: \[128 = 2^7\]. Therefore, the expression becomes: \text{\(\sqrt[6]{\frac{2^7 u^7}{v^3}}\)}.

Step 3: Apply the exponent rules inside the radical

Now apply the quotient rule inside the radical: \[\sqrt[6]{2^7 u^7} \times \sqrt[6]{\frac{1}{v^3}}\].

Step 4: Simplify each part separately

Simplify each part inside the radical separately: \text{\(\sqrt[6]{2^7 u^7} = (2^7 u^7)^{1/6} = 2^{7/6} u^{7/6}\)} and \text{\(\sqrt[6]{\frac{1}{v^3}} = \left( \frac{1}{v^3} \right)^{1/6} = v^{-3/6} = v^{-1/2}\)}.

Step 5: Combine the simplified components

Combine the simplified components: \text{\(2^{7/6} u^{7/6} v^{-1/2}\)}.

Step 6: Express the final simplified form

The final simplified form is \text{\(2^{7/6} u^{7/6} v^{-1/2}\)}.

Key Concepts

Radical Expressions

In mathematics, a radical expression involves roots, such as square roots, cube roots, and other complex roots, notated with the radical symbol \(\root{n}{(...)}\). For example, the square root of 16 is 4, written as \(\root{2}{16} = 4\). Here, we deal with the sixth root, which is expressed as \(\root{6}{(...) }\). When simplifying radical expressions, the goal is to reduce them into their simplest form by factoring and combining like terms.
By understanding how to manipulate these expressions, we can resolve more complex mathematical problems with ease.

Exponent Rules

Exponent rules are fundamental when simplifying radical expressions. These rules include:

• Power of a Power: \(a^{m \times n} = (a^m)^n \)
• Product of Powers: \(a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \(a^m / a^n = a^{m-n} \)

In our example, to simplify \(\root{6}{2^7 u^7} \times \root{6}{\frac{1}{v^3}} \), we use the property \(a ^{\frac{m}{n}} \) to extract roots. This means \(\( \root{n}{a^m} = a^{m/n}\) \). Therefore, \(\( \root{6}{2^7} = 2^{7/6}\) \) and \(\( \root{6}{u^7} = u^{7/6} \)\). Understanding these rules makes simplifying many radical expressions straightforward.

Prime Factorization

Prime factorization steps in to simplify components inside the radical. It involves breaking down a number into its basic prime factors. For example, 128 can be factored into primes as \(2^7 \) (since 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2). In the current example, rewriting 128 as \(2^7 \) allows us to use the exponent rules to simplify it within the radical: \(\root{6}{128} = \root{6}{2^7}\ = 2^{7/6}\). Prime factorization is crucial to simplifying radical expressions as it transforms complex components into manageable and recognizable forms.

Quotient Rule

The quotient rule for exponents is essential when simplifying terms within a fraction under a radical. The rule states that \( \root{n}{\frac{a}{b}} = \frac{\root{n}{a}}{\root{n}{b}} \). Using this rule, we separate \(\root{6}{\frac{2^7 u^7}{v^3}} \) into two parts: the numerator and the denominator. This separation makes each part easier to simplify:

• For the numerator: \(\root{6}{2^7 u^7} = 2^{7/6} u^{7/6} \)
• For the denominator: \(\root{6}{\frac{1}{v^3}} = v^{-1/2} \)

Combining these results we get: \(\frac{2^{7/6} u^{7/6}}{v^{1/2}} \), and finally express the simplified form as \(2^{7/6} u^{7/6} v^{-1/2} \). Understanding the quotient rule helps break down and simplify complex fractions under radicals.