Problem 9 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression in radical notation is \( 2 \sqrt[3]{m} \). It is already simplified.

Step 1: Understand Radical Notation

The expression given is written in exponential notation where the exponent is a fraction: \( m^{1/3} \). The rule for converting from exponential to radical notation is that \( a^{1/n} \) is equivalent to \( \sqrt[n]{a} \). This means the expression \( m^{1/3} \) can be rewritten using radical notation as \( \sqrt[3]{m} \).

Step 2: Apply Radical Notation

Apply the conversion rule from Step 1 to rewrite our expression \( 2 m^{1/3} \). Using the radical notation, this becomes \( 2 \sqrt[3]{m} \).

Step 3: Simplify the Expression

Look for any further simplification in the expression \( 2 \sqrt[3]{m} \). Since there are no particular simplifications possible with this expression in its current form, \( 2 \sqrt[3]{m} \) is already the simplest form.

Key Concepts

Exponential Notation

Exponential notation is a way to express numbers using a base and an exponent, which are very handy in algebra. When you see something like \( m^{1/3} \), this is in exponential notation. It tells us to take the variable \( m \) and raise it to a power of \( 1/3 \). This can be a bit confusing at first, but it's simply a way to express roots and powers concisely.

Here's a simple way to interpret it:

• The base is \( m \).
• The exponent is \( 1/3 \), which is a fractional exponent.

In this context, a fractional exponent like \( 1/3 \) is a signal to find the cubic root of \( m \). In exponential notation, this base-exponent relationship is compact but powerful, allowing us to perform complex operations in a simpler form.

Cubic Root

The cubic root is a special type of root in mathematics, symbolized by \( \sqrt[3]{x} \). It basically asks, "What number multiplied by itself three times gives \( x \)?" In other words, the cubic root of a number \( x \) is a number \( y \) such that \( y^3 = x \).

In the expression \( m^{1/3} \), we are specifically dealing with the cubic root of \( m \). By applying radical notation, we convert \( m^{1/3} \) into \( \sqrt[3]{m} \). This translation is a simplified way of understanding roots, which are fundamental in algebra for simplifying expressions and solving equations.

Remember:

• The cube root of a negative number is also negative as \( (-a)^3 = -a^3 \).
• Cubic roots are applicable to both rational and irrational numbers.

Understanding cubic roots and their notations can ease the complexity of solving various algebraic problems.

Simplification in Algebra

Simplification in algebra is the process of reducing an expression to its most basic form, making it easier to work with or understand. When dealing with expressions such as \( 2 \sqrt[3]{m} \), simplification might involve reducing terms or factoring expressions if possible.

In the exercise given, the expression \( 2 m^{1/3} \) is already quite simplified due to:

• The \( 2 \) is a constant that cannot be simplified further with the \( m \) term.
• \( \sqrt[3]{m} \) is the simplest radical form of \( m^{1/3} \) when rewritten from exponential notation.

Thus, \( 2 \sqrt[3]{m} \) is the simplest because there aren’t any like terms to combine or any additional operations to perform. Understanding simplification is key in algebra as it not only makes expressions easier to interpret but also helps in solving more complex algebraic problems efficiently.