Problem 65 - Algebra 2

Question

Simplify each radical expression. \(n\) is an even number. $$ \sqrt[n]{m^{4 n}} $$

Step-by-Step Solution

Answer

\(m^4\)

Step 1: Identify the powers

First, we should identify the root number \(n\) which is an even number and the power of \(m\) which is \(4n\).

Step 2: Apply the root property

Applying the property of roots that the n-th root of a quantity raised to a n-th power is the original quantity, we get \(m^{4n/n}\).

Step 3: Simplify the exponent

Simplify the exponent by cancelling out \(n\), which leads to \(m^4\).

Key Concepts

Even Number

An even number is any integer that is divisible by 2. These numbers include 0, 2, 4, 6, and so on. The concept of even numbers is crucial because, in mathematics, certain properties and simplifications only apply when dealing with even numbers.

• Even numbers have a remainder of 0 when divided by 2.
• They are symmetric around zero on a number line, meaning both positive and negative even integers are equally spaced around zero.

In the context of radical expressions, knowing that a number is even helps in simplifying roots. For instance, when the root number \( n \) is even, it ensures that the expression inside the radical can be symmetrically and predictably reduced, thus simplifying computations.

Root Property

The root property is fundamental when working with radical expressions. It states that the n-th root of a quantity that is raised to an n-th power equals the absolute value of the base quantity. This property is especially practical when simplifying expressions that include roots.

• If you have \( \sqrt[n]{a^n} \), the result simplifies directly to the absolute value \( |a| \).
• This is because the n-th root and the n-th power are inverse operations, effectively canceling each other out.

Knowing and applying the root property makes it straightforward to simplify complex radical expressions. This occurs in two steps: identifying the radical and exponent, and then applying the property to condense the expression.

Exponents

Exponents represent repeated multiplication. For example, \( m^4 \) means \( m \times m \times m \times m \). They are crucial in reducing lengthy multiplication into a more manageable form.

• Exponents follow specific rules, such as the power of a power rule, which says \( (a^m)^n = a^{m \cdot n} \).
• They also have a direct correlation with roots; when expressions involve exponents and roots, simplification often applies exponent rules to ease calculations.

In our exercise, understanding that \( m^{4n} \) can first be simplified using the n-th root helps to manage and minimize complex algebraic manipulations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often makes the expression easier to work with and understand.

• Breaking down a radial expression is done by factoring, finding common powers, and applying properties like the root property.
• With radical expressions, simplification mainly includes reducing the root and exponent such that no further simplification is possible.
• Steps typically include observing numerator and denominator relationships, like cancelling common terms and using exponent rules.

For instance, in the exercise, simplifying involves reducing \( \sqrt[n]{m^{4n}} \) to \( m^4 \) by applying the root property and cancelling the common factors, which is a key step in understanding and solving complex algebraic issues.