Problem 64 - Algebra 2

Question

Simplify each radical expression. \(n\) is an even number. $$ \sqrt[n]{m^{3 n}} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[n]{m^{3 n}}\) is \(m^3\).

Step 1: Identify the index and the exponent

For the expression \(\sqrt[n]{m^{3 n}}\), \(n\) is the index or the root of the radical and \(3n\) is the exponent of the radicand \(m\).

Step 2: Apply the property of radicals

Using the property of radicals: \(\sqrt[n]{a^n} = a\), our expression can be simplified. As the exponent of the radicand is a multiple of the root of the radical, it fits this property.

Step 3: Simplify the radical

Applying the property of radicals to our expression \(\sqrt[n]{m^{3 n}}\) gives us the simplified form \(m^3\).

Key Concepts

Radical Expressions

When dealing with radical expressions such as \( \sqrt[n]{m^{3n}} \), it's important to understand what a radical is and how to simplify it. Essentially, a radical involves a root symbol, like the square root or cube root. In this case, the expression involves an \(n\)-th root.

Radical expressions contain a radicand (the number or expression under the root symbol) and an index (the small number just above the root symbol indicating the degree of the root). If no index is written, it’s typically a square root with an implied index of 2.

Simplifying radical expressions means rewriting them in a simpler format. You can often use properties of radicals or laws of exponents to do this. In this exercise, understanding how to manipulate roots and exponents is fundamental.

Properties of Radicals

Radicals have several properties that make simplifying expressions easier. One key property is: \( \sqrt[n]{a^n} = a \) for even values of \(n\). This means if the exponent on the radicand matches the root's index, the two cancel out, simplifying the expression to just the base, \(a\).

Here’s another property for simplifying radicals: \( \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \). It allows you to separate the product of two numbers under a single radical into two separate radicals. This is useful when simplifying expressions where the radicand is a product.

Understanding and applying these properties efficiently can dramatically simplify complicated-looking radical expressions, turning them into something much more manageable. As seen in the exercise, using the correct property makes the expression easy to resolve.

Exponents

Exponents tell you how many times a number, known as the base, is multiplied by itself. In the expression \(m^{3n}\), \(m\) is the base, and \(3n\) is the exponent, indicating \(m\) is multiplied by itself \(3n\) times.

Key rules about exponents help in simplifying expressions with radicals. The power of a power rule states \((a^m)^n = a^{mn}\). This is particularly useful in simplifying complex expressions involving both exponents and radicals.

Understanding the relationship between exponents and radicals is crucial, since radicals can often be expressed using fractional exponents. For example, \(\sqrt[n]{a} \) can be rewritten as \(a^{1/n}\). In our exercise, recognizing that \(\sqrt[n]{m^{3n}} \) can be simplified using exponent rules was crucial to reaching the answer, turning it into \(m^3\). This interplay between roots and exponents increases the versatility and power of algebraic manipulation.