Problem 52 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[2 n]{x^{2 n}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[2n]{x^{2n}} \) is \( |x| \).

Step 1: Apply exponent rules

In this step, we should recognize that, by the power of a power property, we can simplify the expression by dividing the exponent of x (which is 2n) by the root (which is also 2n). So, the simplified version of \(x^{2n}\) under a \(2n\) root is \(x\). So, we have: \( \sqrt[2n]{x^{2n}} = x \)

Step 2: Consider absolute values

However, we have to consider the possibility of \(x\) being negative. If \(x\) is negative, then a negative number to an even power would be positive. To keep the value correct, we have to take the absolute value of \(x\). Hence, the final simplified expression is \( |x| \).

Key Concepts

Exponent Rules

Exponent rules are fundamental in simplifying expressions that involve powers and roots. One key rule is the property that states when you multiply powers with the same base, you simply add their exponents. Another important rule is the "power of a power" property, which we'll cover more depth later.

• **Multiplying Powers**: For any base \(a\) and exponents \(m\) and \(n\), \(a^m \times a^n = a^{m+n}\).
• **Dividing Powers**: When dividing powers of the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• **Negative Exponents**: A negative exponent \(a^{-n}\) is the same as \(1/a^n\).

Understanding these rules is crucial in breaking down more complex expressions into manageable parts.

Absolute Value

Absolute value represents the distance of a number from zero on a number line, essentially making any number positive. In mathematical terms, the absolute value of a number \(x\), denoted \(|x|\), is:

• \(|x| = x\) if \(x\) is positive or zero,
• \(|x| = -x\) if \(x\) is negative.

The importance of absolute value arises when dealing with roots of even degree. Since any real number raised to an even power is positive, we use the absolute value to ensure the expression remains non-negative, regardless of whether the original input was positive or negative. This is why in the expression \( \sqrt[2n]{x^{2n}} \), the simplified form is \(|x|\), to account for potential negative values of \(x\).

Power of a Power Property

The "power of a power" property is a specific exponent rule that simplifies expressions wherein an exponent is raised to another exponent. According to this property, for any base \(a\) and exponents \(m\) and \(n\), raising a power to another power is accomplished by multiplying the exponents together: \[ (a^m)^n = a^{m \times n} \]This property becomes particularly useful when simplifying radical expressions. For instance, in the exercise \( \sqrt[2n]{x^{2n}} \), by employing the power of a power property, you recognize that the exponent inside the root (which is \(2n\)) can be divided by the degree of the root (also \(2n\)), resulting in \(x^{(2n)/(2n)} = x\). This step is pivotal before applying any additional rules like considering absolute values, leading to the final expression of \( |x| \).