Problem 73 - Algebra A Combined Function

Question

Assume that all variables represent positive real numbers. $$ \sqrt[5]{-32 x^{10} y^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x^2y\).

Step 1: Identify the Expression Inside the Radical

We need to simplify the expression \( \sqrt[5]{-32 x^{10} y^{5}} \). Notice that the expression inside includes the components \(-32\), \(x^{10}\), and \(y^{5}\).

Step 2: Simplify the Numerical Part

-32 is equivalent to \(-1 \times 32\). We can rewrite \(32\) as \(2^5\). Therefore, \(-32 = -1 \times 2^5\).

Step 3: Simplify the Power of Variables

The term \(x^{10}\) can be rewritten as \((x^2)^5\), and \(y^5\) is already in the power of 5. These are perfect fifth powers.

Step 4: Apply the Fifth Root

The fifth root of each component is calculated: \( \sqrt[5]{-1 \times 2^5 \times (x^{2})^5 \times y^5} = \sqrt[5]{-1} \times 2 \times x^2 \times y\).

Step 5: Evaluate the Roots

Since the fifth root of \(-1\) is \(-1\) (because an odd root of a negative number remains negative), the final expression is simplified as \(-2x^2y\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or fifth roots. They allow you to "undo" powers by finding numbers that, when raised to a specific power, return the original number. Radical expressions are often used to simplify algebraic equations and solve for unknown variables.
For example, if you have the expression \( \sqrt{16} \), you are looking for a number that, when squared, equals 16. Therefore, \( \sqrt{16} = 4 \), because \( 4^2 = 16 \).

• The symbol \( \sqrt{} \) represents the square root.
• Higher roots are indicated with a small number above the radical sign, such as \( \sqrt[5]{} \) for the fifth root.
• Radicals can also involve variables, such as \( \sqrt{x} \) or \( \sqrt[3]{x^3} \).

When dealing with radicals, remember that they are inverses of exponential functions. They help simplify expressions where numbers are raised to a power, bringing them back to their original or reduced form.

Fifth Roots

The fifth root of a number is a value that, when raised to the power of 5, results in the original number. For instance, the fifth root of 32 is 2, because \( 2^5 = 32 \). The notation for fifth roots is typically written as \( \sqrt[5]{x} \).
Fifth roots are particularly interesting because they allow us to work with negative numbers while staying within the domain of real numbers, assuming that the expression is simplified correctly. This is because odd roots like the fifth root maintain the sign of the base number. Thus, \( \sqrt[5]{-32} = -2 \), as \( (-2)^5 = -32 \).

• Fifth roots help in simplifying expressions by reducing them to their basic components.
• They retain the characteristic of the base number being positive or negative depending on the circumstances.
• They simplify powers of variables such as \( x^{10} \) by reducing them to \( (x^2)^5 \).

It's key to understand fifth roots to tackle problems involving higher powers efficiently and to appreciate the subtleties of algebraic manipulation.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This makes calculations easier and reveals any underlying patterns or relationships more clearly. Simplification often involves combining like terms, canceling out factors, and using properties of exponents and roots.
In the example \( \sqrt[5]{-32 x^{10} y^{5}} \), we simplify by breaking down each part within the radical. We see that we can rewrite numeric expressions like \(-32\) as \(-1 \times 2^5\), and the powers of variables, such as \(x^{10}\) as \((x^2)^5\).

• First, address the numerical part of the expression. Transform numbers into exponential form that matches the root, like \(2^5\) in this case.
• Transform variable expressions to have their exponents match the root degree.
• Apply the radical to each part of the expression separately and simplify the expression step-by-step.

The goal in simplifying expressions is to rewrite them in the most straightforward form possible. This usually involves breaking them down into identifiable components and then using algebraic rules to condense them.

Positive Real Numbers

In algebra, we often deal with positive real numbers, which are all the numbers greater than zero on the number line. This concept forms the foundation for many algebraic rules and operations, especially with radicals and powers.
Positive real numbers behave predictably with respect to operations like addition, multiplication, and taking roots. Specifically, taking an even root (like a square root) of a positive number results in a positive number. However, taking an odd root (like a cube root or fifth root) allows for positive and negative outcomes based on the sign of the original number.

• Positive real numbers ensure that operations within radicals are well-defined and avoid ambiguous results.
• They allow us to use expressions without worrying about undefined or complex numbers emerging during simplification processes.
• Understanding the behavior of positive real numbers is crucial when simplifying expressions to maintain their validity and real characteristics.

Ultimately, operations on positive real numbers provide a consistent framework that is indispensable in solving algebraic problems efficiently.