Problem 10 - Intermediate Algebra

Question

Use radical notation to write each expression. Simplify if possible. $$ (2 m)^{1 / 3} $$

Step-by-Step Solution

Answer

\(\sqrt[3]{2m}\)

Step 1: Identify the Expression

The given expression is \((2m)^{1/3}\). This expression involves taking the cube root of \(2m\).

Step 2: Convert to Radical Notation

In radical notation, the expression \((2m)^{1/3}\) is represented as \(\sqrt[3]{2m}\). This means we take the cube root of the entire expression inside the parenthesis.

Step 3: Simplify if Possible

Check if the expression inside the radical can be simplified. Since \(2m\) does not have any cube factors that can be taken out, no simplification is possible for \(\sqrt[3]{2m}\). Thus, the expression in radical notation remains as \(\sqrt[3]{2m}\).

Key Concepts

Cube Root

The cube root of a number is a special type of root operation. While a square root asks "what number multiplied by itself will equal the given number?", a cube root asks "what number multiplied by itself twice will result in the given number?".
Cube roots are written in radical notation using the radical sign with the index of 3, for example \( \sqrt[3]{x} \). This simply means find the number which, when raised to the power of three, equals \( x \).
To find the cube root of a number or expression in algebra, we often start with an expression in exponential form, such as \( a^{1/3} \). Converting this into radical notation gives us \( \sqrt[3]{a} \). Understanding cube roots is essential as they often simplify expressions, especially in algebra and geometry.

Exponent Rules

Exponent rules are fundamental when working with algebraic expressions, especially when converting expressions between exponential and radical forms. One of the key rules is that the exponent \(1/n\) means we are taking the \(n\)-th root of a number.
Thus, an expression like \( (2m)^{1/3} \) suggests taking the cube root. The basic exponent rules include:

• \( a^{m} \cdot a^{n} = a^{m+n} \)
• \( (a^{m})^{n} = a^{m \cdot n} \)
• \( a^{-n} = \frac{1}{a^{n}} \)

Applying these rules allows us to simplify and manipulate complex algebraic expressions effectively. These rules are tools that make expressions easier to work with, helping us to rearrange and break down expressions in algebra.

Simplification

Simplification in algebra involves reducing an expression to its simplest form. It's about making an expression as compact and straightforward as possible. When simplifying an expression involving roots, like a cube root, we check to see if there are factors within the expression that can be simplified further.
For instance, in \( \sqrt[3]{2m} \), we would look at both \(2\) and \(m\) to determine if either has a cube that can be extracted. This might not always be possible, as in the example, where neither component of the expression has a cube factor.
When simplification doesn’t yield a different result, it reinforces understanding that an expression is already as neat as it can be. Simplification helps in making calculations more manageable in further mathematical operations.

Radicals

Radicals are expressions that involve roots—such as square roots, cube roots, or higher-order roots. They are denoted by the radical symbol \(\sqrt{}\), and often include an index to specify which root is being taken, such as \(\sqrt[3]{x} \) for cube roots.
Understanding and operating with radicals involve knowing how to interpret radical expressions and convert between radical and exponential forms. This conversion is useful because some mathematical operations, such as multiplication and division, might be easier to complete when using exponentials rather than radicals.
For example, it's often easier to work with \(a^{1/3} \cdot b^{1/3}\) as \((ab)^{1/3} \) rather than \( \sqrt[3]{a} \cdot \sqrt[3]{b} \). Radicals can also be simplified in cases where the radicand (the value inside the radical symbol) has perfect square or cube factors. Mastery of radicals is crucial in algebra to handle different kinds of equations efficiently.