Problem 54 - College Algebra with Modeling and Visualization

Question

Simplify the radical expression by factoring out the largest perfect nth power. Assume that all variables are positive. $$ \sqrt[5]{\frac{4 t^{6}}{r}} \cdot \sqrt[5]{\frac{8 t}{r^{6}}} $$

Step-by-Step Solution

Answer

\( \frac{2t^{6/5} \cdot \sqrt[5]{4t^2}}{r^{6/5}} \)

Step 1: Simplify Each Radical Expression Separately

Begin by simplifying each radical expression separately. The first expression is \( \sqrt[5]{\frac{4t^6}{r}} \). The second expression is \( \sqrt[5]{\frac{8t}{r^6}} \). We will simplify within each radical.

Step 2: Factor Out Largest Perfect Powers

Identify perfect fifth powers within the radicals. In the first expression, \( t^6 \) can be written as \( t^5 \cdot t \). In the second, \( r^6 \) can be written as \( r^5 \cdot r \), and 8 can be split into a perfect fifth power by using its factors, \( 8 = 2^3 \).

Step 3: Rewrite and Simplify the Radicals

Rewrite the expressions factoring out the largest fifth power. The first radical becomes \( \sqrt[5]{\frac{4t^6}{r}} = \sqrt[5]{\frac{t^5 \cdot 4t}{r}} \). The second becomes \( \sqrt[5]{\frac{8t}{r^6}} = \sqrt[5]{\frac{2^3 t}{r^5 r}} \).

Step 4: Simplify Individual Radicals

Use the property \( \sqrt[5]{a^5} = a \). The first becomes \( \frac{t}{r^{1/5}} \cdot \sqrt[5]{4t} \), and the second becomes \( \frac{2t^{1/5}}{r} \cdot \sqrt[5]{t} \).

Step 5: Multiply the Simplified Expressions

Multiply the simplified expressions: \( \left(\frac{t}{r^{1/5}} \cdot \sqrt[5]{4t}\right) \cdot \left(\frac{2t^{1/5}}{r} \cdot \sqrt[5]{t}\right) \).

Step 6: Simplify the Product

Combine the terms: \( \frac{2t^{1+1/5} \cdot \sqrt[5]{4t^2}}{r^{1/5 + 1}} = \frac{2t^{6/5} \cdot \sqrt[5]{4t^2}}{r^{6/5}} \).

Step 7: Final Simplification

Combine powers of \( t \) and \( r \): \( \frac{2t^{6/5} \cdot \sqrt[5]{4t^2}}{r^{6/5}} \). This is the final simplified form of the expression.

Key Concepts

Simplifying Radicals

Simplifying radicals is a crucial skill in algebra that involves reducing radical expressions to their simplest form. A radical expression contains a radical sign (√) and includes numbers or variables. The goal of simplifying radicals is to make them easier to work with by reducing the components under the radical to their simplest values.
To simplify a radical expression:

• First, identify and extract the largest possible perfect power from underneath the radical sign. This is important because it allows you to simplify the expression more effectively.
• The properties of radicals state that multiplying the same root of two terms allows you to combine them under a single radical.
• It's important to note, when simplifying higher-order roots, that you have to identify perfect powers like cubes, fourths, and fifths.

Let's use a fifth root as an example: To simplify \( \sqrt[5]{32} \), recognize that 32 can be expressed as \( 2^5 \), which is a perfect power. Thus, the expression can be simplified to the integer 2.
By applying these principles to more complex expressions, you can handle various challenging algebraic tasks more smoothly.

Perfect nth Powers

A perfect nth power is a number that can be expressed as the integer 'a' raised to the power 'n', such as \( a^n \). These numbers are crucial in simplifying radical expressions because they allow you to take roots easily.
When simplifying radical expressions involving higher roots, identifying the perfect nth powers makes the process efficient:

• They help in breaking down complex expressions so they are more manageable.
• For example, in the expression \( \sqrt[5]{t^6} \), recognizing \( t^6 \) can be rewritten as \( t^5 \cdot t \) helps to simplify it, since \( \sqrt[5]{t^5} = t \).
• This method emphasizes factoring out perfect powers from variables and constants alike.

Recognizing these perfect nth powers is not only vital for simplification but also plays a big role in solving equations where radicals are involved. Once you've factored out the largest perfect power, the leftover factor can continue to simplify the expression.

Rational Exponents

Rational exponents are another way to express radicals and roots, providing a bridge between exponential and radical notation. A rational exponent is written as a fraction, and it denotes both an exponent and a root. For example, \( x^{m/n} \) translates to the nth root of \( x^m \).
Understanding this concept is essential:

• Ironically, rational exponents often make expressions simpler to work with than their radical counterparts because they follow the rules of exponents.
• Using rational exponents, you can easily manipulate expressions using multiplication and division properties that apply to exponents.
• For example, \( x^{1/2} \cdot x^{1/3} \) simplifies to \( x^{(1/2 + 1/3)} = x^{5/6} \).

Rational exponents are particularly handy when you need to standardize expressions for operations such as differentiation or integration. Knowing how to toggle between radical form and rational exponents expands your algebraic toolbox and empowers you to tackle expressions more confidently.