Problem 37 - Technical Mathematics with Calculus

Question

Simplify. $$3 \sqrt[5]{32 x y^{11}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(6y^2 \root{5}{x y}\).

Step 1: Identify the Perfect Fifth Powers

First, identify any numbers inside the radical that are perfect fifth powers. In this case, 32 can be expressed as \(2^5\). Also, factor out the highest power of y that is a multiple of 5, which is \(y^{10} = (y^2)^5\). The expression becomes \(3 \times 2 \times y^2 \times \root{5}{x y}\).

Step 2: Simplify the Radical

Simplify the expression by multiplying the coefficients outside the radical and apply the fifth root to the perfect fifth powers inside the radical. Thus, \(3 \times 2 \times y^2\) becomes \(6y^2\) and the expression simplifies to \(6y^2 \root{5}{x y}\).

Step 3: Rewrite the Simplified Expression

Combine all the simplified components to get the final simplified expression: \(6y^2 \root{5}{x y}\). This is the simplified form of the original expression.

Key Concepts

Perfect Fifth Powers

When working with radical expressions, particularly roots higher than the common square or cube roots, understanding perfect fifth powers is essential. A perfect fifth power is a number that can be expressed as another number raised to the power of five, such as \( 32 = 2^5 \). In simplifying radicals, it's crucial to identify these perfect fifth powers because they can be taken out of the radical, making the expression easier to handle.

Here's a quick list of the first few perfect fifth powers to aid in recognition:

• \( 1^5 = 1 \)
• \( 2^5 = 32 \)
• \( 3^5 = 243 \)
• \( 4^5 = 1024 \)
• \( 5^5 = 3125 \)

Being familiar with these numbers allows students to more quickly identify when a component of a radical expression is a perfect fifth power. For instance, in the exercise \(3 \sqrt[5]{32 x y^{11}}\), recognizing that 32 is a perfect fifth power leads to a significant simplification.

Radical Expressions

Radical expressions involve roots, such as square roots \(\sqrt{}\), cube roots \(\sqrt[3]{}\), and in our case, fifth roots \(\sqrt[5]{}\) of numbers. These expressions can look intimidating, but they follow rules similar to regular arithmetic. The key to understanding radicals is realizing that they're the opposite operation of raising a number to a power. For instance, as \(2^5 = 32\), we know that \(\sqrt[5]{32} = 2\).

The complexity increases as variables get involved alongside constants inside the radical. In our example, the expression includes not just a constant but also the variables \(x\) and \((y^{11})\). Simplifying radical expressions often involves factoring out perfect powers, as shown in the provided solution, to reduce the radicals to their simplest form.

Simplification Process

The process of simplifying radicals is a systematic approach to making the expression more manageable. The idea is to remove as much from under the radical as possible by factoring out perfect powers. In our example, the fifth root of 32 is simplified by recognizing 32 as a perfect fifth power of 2, which can then be taken out of the radical.

To further simplify, one can also factor out the highest power of any variable that is a multiple of the index of the radical—in this case, 5. By reducing the exponent of \(y^{11}\) to \(y^{10} \times y\), \(y^{10}\) is factored out of the radical because it is a perfect fifth power of \(y^2\). The remaining \(y\), which is not a perfect power, stays within the radical.

Combining these simplified components outside of the radical yields the final expression, which is easier to use in further calculations or applications. Simplification can be applied in many areas, including geometry, algebra, and calculus, making it a vital skill in mathematics.