Problem 2 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ 64^{1 / 3} $$

Step-by-Step Solution

Answer

The expression \(64^{1/3}\) simplifies to 4.

Step 1: Understand the Expression

The given expression is \(64^{1/3}\). This is written in exponential form where the base is 64 and the exponent is \( \frac{1}{3} \), which indicates a cube root.

Step 2: Convert to Radical Notation

To convert an expression in the form \(a^{m/n}\) to radical notation, we use \( \sqrt[n]{a^m} \). In this case, \(64^{1/3}\) can be written as \( \sqrt[3]{64} \), which means the cube root of 64.

Step 3: Simplify the Radical Expression

Now, we need to find the cube root of 64. Since \(4^3 = 4 \times 4 \times 4 = 64\), the cube root of 64 is 4. Thus, \( \sqrt[3]{64} = 4 \).

Key Concepts

Cube Root

The cube root is a fundamental concept in mathematics. It is the inverse operation of raising a number to the third power. Just like a square root finds a number that, when multiplied by itself, gives the original number, a cube root finds a number that, when cubed, gives the original number.
For example, the cube root of 64, denoted as \( \sqrt[3]{64} \), is 4. This is because \( 4^3 = 64 \).

• When you see the notation \( \sqrt[3]{a} \), it refers to the cube root of \( a \).
• This concept is crucial for solving equations involving cubes and is extensively used in algebra and geometry.

Understanding cube roots helps in simplifying expressions and solving cubic equations.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form. The goal is to remove any perfect cube (or square, depending on the root you are dealing with) from inside the radical sign.
In the expression \( \sqrt[3]{64} \), simplifying involves recognizing that 64 can be expressed as the cube of 4, since \(4^3 = 64\).

• This allows us to simplify \( \sqrt[3]{64} \) to the number 4, because we are effectively taking the cube root of a perfect cube.
• Through this process, we make the expression easier to work with and understand.

Simplifying radicals is a valuable skill in algebra as it reduces complexity in equations and expressions.

Exponential Form

Exponential form is a way of expressing numbers using bases and exponents. It is compact and ideal for representing large numbers compactly.
The expression \( 64^{1/3} \) is presented in exponential form where 64 is the base and \( \frac{1}{3} \) is the exponent. This exponent indicates the cube root operation.

• The general form \( a^{m/n} \) represents the \( n \)-th root of \( a \) raised to the \( m \)-th power. In our example, \( m = 1 \) and \( n = 3 \), simplifying to a cube root.
• Exponential form is especially useful in scientific notation, allowing easier handling of powers and roots.
• Converting between exponential form and radical form helps students understand different mathematical perspectives and operations.

Mastering exponential notation is crucial for higher-level math, including calculus and physics.