Question

\(\sqrt[3]{64 r^3 t^6}\)

Short answer

The simplified form of \(\sqrt[3]{64 r^3 t^6}\) is \(4rt^2\).

Step-by-step solution

Identify the perfect cubes

The first step is to break down each term inside the cube root into cubes. 64 is a perfect cube (4^3), \(r^3\) is obviously a cube, and \(t^6\) can be expressed as \((t^2)^3\), which is a perfect cube as well.

Take the cube root of each term

We can take the cube root of each expression independently thanks to the property of radicals that \(\sqrt[n]{ab} = \sqrt[n]{a} * \sqrt[n]{b}\). Cube root of 64 is 4. Cube root of \(r^3\) is r. Cube root of \(t^6\) (or \((t^2)^3\)) is \(t^2\).

Combine the terms

Finally, bring together all the terms outside the radical. This leaves us with the simplified expression.

Key concepts

Perfect Cubes

Perfect cubes are numbers or expressions that are the result of raising a base number to the power of three. Identifying perfect cubes is crucial to simplifying expressions involving cube roots. For example, in the expression \(64 r^3 t^6\), 64 can be expressed as \(4^3\). Here:

• 64 as a perfect cube: 64 equals \(4 imes 4 imes 4\), which is \(4^3\).
• \(r^3\) is evidently a perfect cube: It's a variable raised to the power of three.
• \(t^6\) as a perfect cube: This can be rewritten as \((t^2)^3\), clearly showing it as a cube.

Recognizing these allows us to simplify cube roots more effectively. The goal in simplifying cube roots is to express them in simpler, whole number forms by breaking them into perfect cubes.

Radicals Simplification

Simplifying radicals, such as cube roots, involves using properties of radicals to express them in their simplest form. For cube roots, one key property is that the cube root of a product can be broken down into the product of the cube roots. This property is stated mathematically as: \(\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}\)In the example given \(\sqrt[3]{64 r^3 t^6}\), each component inside the cube root is a perfect cube:

• Cube root of 64 is 4 because \(4^3 = 64\).
• Cube root of \(r^3\) is \(r\) since \(r^3 / 3 = r\).
• Cube root of \((t^2)^3\) is \(t^2\) as \( (t^2)^{3/3} = t^2\).

By simplifying each of these components separately, we make the overall expression much simpler and easier to work with.

Properties of Exponents

Understanding properties of exponents is essential when simplifying expressions with variables or numbers raised to powers.
These properties make it easier to break down and manage complex expressions, especially when dealing with radicals. Three important properties often used are:

• Power of a power: \((a^m)^n = a^{m \cdot n}\)
• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\)

In our example, the term \(t^6\) was rewritten as \((t^2)^3\). By applying the 'power of a power' property, we take the cube root smoothly as \((t^2)^{3/3} = t^2\).
This technique simplifies complex radical expressions and makes working with them more straightforward.