Problem 41 - Algebra A Combined Function

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[5]{x^{20}} $$

Step-by-Step Solution

Answer

The root is \(x^4\).

Step 1: Understand the Problem

We need to find the fifth root of \(x^{20}\). The goal is to express this radical expression in its simplest form.

Step 2: Apply the nth Root Property

Recall that the \(n\)-th root property states that \(\sqrt[n]{a^m} = a^{m/n}\). Applying this, we can transform \(\sqrt[5]{x^{20}}\) into \(x^{20/5}\).

Step 3: Simplify the Exponent

Simplify \(x^{20/5}\). By dividing the exponent 20 by the root 5, we get \(x^4\).

Step 4: Verify the Solution

Recheck the simplification: The expression \(x^{20/5}\) simplifies to \(x^4\), confirming that the calculations are correct.

Key Concepts

Radical Expressions

Radical expressions involve roots, including the more familiar square roots, and other types like cube roots, fourth roots, and in this case, a fifth root. When you see something like \( \sqrt[n]{a} \), it means you're looking for a number that, when raised to the power \( n \), gives you \( a \). The number \( n \) is called the index of the radical, and \( a \) is the radicand. In the given problem, \( \sqrt[5]{x^{20}} \) can be thought of as asking, "What number multiplied by itself five times equals \( x^{20} \)?" This is where simplifying radical expressions becomes crucial, and it is helpful to understand the transformation of roots into fractional exponents to solve these expressions more easily.

Simplifying Exponents

Simplifying exponents means representing an expression with powers in a less complicated form without altering its value. In the step-by-step solution, the transformation from \( \sqrt[5]{x^{20}} \) to \( x^{20/5} \) is a key example of this. The process begins by applying the rule of exponents for radicals, where the index becomes the denominator and the power of the radicand becomes the numerator. It's important to understand that this fraction represents the division of exponents: \( 20/5 \). By dividing, you reduce \( x^{20/5} \) to \( x^4 \), the simplest possible form for this expression. Each exponentiation simplification requires careful examination to ensure no mistakes are made during reduction.

Exponentiation Rules

Exponentiation rules are the backbone for working with powers in algebra. These rules help with multiplying, dividing, and simplifying expressions with exponents. For example, when simplifying \( \sqrt[5]{x^{20}} \), we use the power of a power rule: \((a^m)^n = a^{m \times n}\). Additionally, knowing that \( \sqrt[n]{a^m} = a^{m/n} \) guides the conversion of radical expressions to exponents, as we saw earlier with \( x^{20/5} \). When multiply exponents, add them; when divide, subtract. Always adjust based on the rules, ensuring the use of correct rules for multiplying and dividing powers. These rules streamline calculations, making complex expressions more manageable and ensuring accurate results.