Problem 54 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[2 n]{x^{6 n}} $$

Step-by-Step Solution

Answer

The simplified form of the radical expression \( \sqrt[2n]{x^{6n}} \) is \( x^{3} \)

Step 1: Simplify the Expression

Use the property of radicals and simplify the expression \( \sqrt[2n]{x^{6n}} \) as \( x^{6n/2n} \).

Step 2: Simplify the Exponent

Simplify \( 6n/2n \) to 3

Step 3: Insert the Simplified Exponent

Replace the exponent in the expression, thus obtaining \( x^{3} \)

Key Concepts

Simplification of Expressions

Simplification of expressions, especially with radicals, is the process of expressing the expression in its simplest form. By simplifying, we aim to make the expression easier to work with while preserving its original value. To simplify the given expression \( \sqrt[2n]{x^{6n}} \), we use the property that \( \sqrt[b]{a^b} = a \). Essentially, this rule helps to "cancel out" the root with the exponent when they are the same, or to simply reduce the power if they are different.

In our exercise, we begin by expressing the radical \( \sqrt[2n]{x^{6n}} \) as a power: \( x^{6n/(2n)} \). This transformation enables us to simplify the fraction \( 6n/2n \), as exponents inside a radical can be treated like fractions. Ultimately, this leads to a more manageable expression, which in this case is \( x^3 \). This simplified form is much easier to work with in further calculations or in solving equations.

• Simplifying radicals often involves converting the root into fractional exponents.
• Factor the expression to identify powers that can be "canceled out" by the radical.
• Reform and reduce expressions using exponent rules to find a simpler equivalent form.

Absolute Value

Absolute value is a concept that refers to the non-negative value of a number without regard to its sign. When working with radical expressions, particularly when simplifying them, it is crucial to consider whether the result could potentially take on negative values.

In the context of our example, \( x^3 \) might initially seem straightforward; however, the absolute value consideration ensures we account for all possible values of \( x \). If \( n \) is an even number, the absolute value becomes relevant because \( \sqrt[2n]{x^{6n}} \) was originally a form that suggests non-negative outcomes—as roots of even degrees do.

• Absolute value helps in ensuring the output adheres to expected properties of radicals, like always being non-negative.
• It is crucial when the variable \( x \) can take both positive and negative values.
• For even roots, always check if the absolute value symbols are needed due to possible sign changes.

Exponent Rules

Exponent rules are fundamental in simplifying expressions with powers and radicals. Exponents dictate how many times a number is to be multiplied by itself, and understanding these rules unlocks the path to simplifying complex expressions.

When simplifying expressions like \( \sqrt[2n]{x^{6n}} \), we rely on the rule that \((a^m)^n = a^{m \cdot n}\), allowing us to effectively manipulate the powers. For example, the expression \( x^{6n} \) over a 2n-th root simplifies by dividing exponents as we initially set \( x^{6n/(2n)} \). This utilizes another rule, known as the power of a power rule, where despite the existence of a fractional or nested exponent, the simplification directly follows through a division of exponents.

• Exponents within radicals require careful simplification using division and multiplication rules.
• Be attentive to how exponents multiply when nested inside roots.
• Use exponent rules to transform complex expressions into simpler ones, facilitating easier calculations.