Problem 35 - Intermediate Algebra

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 Expression

The expression given is \( \sqrt[5]{x^{20}} \). This represents the fifth root of \( x^{20} \). Our goal is to simplify this expression.

Step 2: Apply the Root and Power Rule

According to the rule that \( \sqrt[n]{a^m} = a^{m/n} \), we can rewrite the expression \( \sqrt[5]{x^{20}} \) as \( x^{20/5} \).

Step 3: Simplify the Exponent

Simplify the fraction \( 20/5 \) which results in \( 4 \). Therefore, the expression simplifies to \( x^4 \).

Step 4: Conclude the Simplification

Now that we have \( x^4 \), which implies that the fifth root of \( x^{20} \) is \( x^4 \). Since we are assuming nonnegative real numbers, this is the required root.

Key Concepts

Fifth Root

Understanding what a fifth root is can be straightforward if you know about square roots or cube roots. Essentially, the fifth root of a number is a value that, when multiplied by itself four more times (a total of five times), gives the original number. In mathematical terms, if you have a number or an expression like \( x \), then the fifth root of \( x \) is written as \( \sqrt[5]{x} \).

• For example, to find the fifth root of \( 32 \), you need a number \( a \) such that \( a^5 = 32 \). The answer is \( 2 \), because \( 2^5 = 32 \).
• If the number contains a variable with an exponent, like \( x^{20} \), finding the fifth root involves dividing the exponent by 5.

This means that the operation essentially reduces the power of the expression from twenty down to four in our example because \( 20 \div 5 = 4 \). That's why \( \sqrt[5]{x^{20}} = x^4 \).

Nonnegative Real Numbers

The concept of nonnegative real numbers is important in mathematics. These are numbers that are either positive or zero. In contrast to negative numbers, nonnegative numbers do not go below zero.
Using nonnegative real numbers ensures that when you evaluate roots, results don't become undefined or involve imaginary numbers. For square and even higher evens roots, dealing with negative numbers could often lead to undefined expressions in real number terms.

• For example, the fifth root and any odd root can indeed be applied to both positive and negative numbers without running into problems, but here we stick to the nonnegative case for consistency and simplicity.
• In our expression \( \sqrt[5]{x^{20}} \), assuming \( x \) is a nonnegative real number means \( x \geq 0 \), making our root result clear and simple.

Assuring the numbers are nonnegative is a crucial step when simplifying expressions with roots.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that allows you to rewrite expressions in a more manageable form without changing their value. This often involves reducing exponents or combining like terms.
When dealing with roots, especially the fifth root as in the example \( \sqrt[5]{x^{20}} \), simplifying becomes a process of rewriting the root in terms of exponents using known rules.

• The rule \( \sqrt[n]{a^m} = a^{m/n} \) helps in rewriting roots in terms of exponents so you can directly perform the operation of division indicated by the denominator of the root.
• By doing so, \( \sqrt[5]{x^{20}} \) becomes \( x^{20/5} \), which simplifies easily to \( x^4 \) after performing the arithmetic division.

This efficient method removes complexity, making it easier to work with expressions and understand underlying patterns. By simplifying, you don't only find answers more quickly, but also build a deeper understanding of how exponents and roots interact.