Problem 48 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{k^{15}} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[5]{k^{15}}\) is \(k^3\) for \(k > 0\) and \(|k^3|\) for \(k < 0\).

Step 1: Understand the expression

The given expression is the fifth root of \(k^{15}\), which is written as \(\sqrt[5]{k^{15}}\). This means we are looking for a number which, when raised to the power of 5, gives \(k^{15}\).

Step 2: Simplify root expression using rules of exponents

We know that \(\sqrt[n]{x^n}\) is equal to \(x\) for \(x > 0\) and \(|x|\) for \(x < 0\). Hence, \(\sqrt[5]{k^{15}}\) equals \(k^3\) for \(k > 0\) and \(|k^3|\) for \(k < 0\).

Key Concepts

Fifth Root

Exploring the concept of the fifth root involves understanding the idea of reversing exponentiation. When we talk about a fifth root, we ask what number needs to be raised to the fifth power to yield a certain value.
For example, \(\sqrt[5]{32} = 2\) because \(2^5 = 32\). The notation \(\sqrt[5]{x}\) represents the fifth root of \(x\).

• The fifth root can be thought of as the number which, when multiplied by itself four more times (a total of five times), results in the original value.
• This concept applies similarly to any other roots, such as square roots, cube roots, etc., by altering the number of times the multiplication occurs.

These roots allow us to simplify expressions where values are raised to higher powers by finding an expression or a number that, when raised to the specified power, regenerates the initial quantity.

Rules of Exponents

The rules of exponents, sometimes called laws of exponents, help simplify expressions where variables are raised to powers. These rules guide us in manipulating these powers efficiently.

• Multiplication Rule: When multiplying powers with the same base, \(a^m \times a^n = a^{m+n}\).
• Division Rule: When dividing powers with the same base, \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: When raising a power to another power, \( (a^m)^n = a^{m \times n} \).
• Root of a Power: \(\sqrt[n]{a^m} = a^{m/n}\).

In the given exercise \(\sqrt[5]{k^{15}}\), this last rule, "root of a power", is applied, as \(k^{15/5} = k^3\). This rule helps transform radical expressions into exponential forms, simplifying calculations.

Absolute Value

Absolute value is a mathematical function that denotes the non-negative value of a number without regard to its sign. It's usually represented with vertical bars like \(|x|\). For any real number \(x\):

• if \(x \geq 0\), then \(|x| = x \).
• if \(x < 0\), then \(|x| = -x \).

This is crucial when simplifying radical expressions, especially when dealing with even roots.
In the context of the exercise, if \(k\) is negative, the output of the root should reflect positivity for real numbers, ensuring the simplified expression accounts for this by using absolute value symbols: \(|k^3|\). Absolute value ensures outcomes align mathematically and practically when dealing with roots.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form. This often means converting radical expressions into exponential terms using the rules of exponents.
For example, to simplify \(\sqrt[5]{k^{15}}\), we apply the roots of powers rule: \(k^{15/5} = k^3\). If the base \(k\) is negative, the absolute value \(|k^3|\) ensures the expression is simplified correctly without losing or inaccurately transforming the expression's numerical value.Follow these steps in general to simplify:

• Convert the radical to an exponent: \(\sqrt[n]{x^m} = x^{m/n}\).
• Simplify the exponent using laws of exponents.
• Ensure the result is written in its simplest form.

This approach demystifies what can appear initially complex, making radicals easier to work with and understand through step-by-step simplification processes.