Problem 17 - Algebra 2

Question

Multiply and simplify. Assume that all variables are positive. $$ \sqrt[3]{6} \cdot \sqrt[3]{16} $$

Step-by-Step Solution

Answer

\sqrt[3]{96}

Step 1: Multiply under cube roots

Multiply the two numbers 6 and 16 which are under the cube roots. So, our new expression becomes \( \sqrt[3]{6 \times 16} \).

Step 2: Simplification

Compute the multiplication under the cube root. This leads to \( \sqrt[3]{96} \).

Step 3: Final Simplification

Simplify the cube root. Since 96 is not a perfect cube, the expression \( \sqrt[3]{96} \) is the simplest form and can not be simplified more.

Key Concepts

Cube Roots

A cube root is a number that, when multiplied by itself three times, gives the original number. In mathematical terms, if you have a number \( x \), then the cube root of \( x \) is a number \( y \) such that \( y^3 = x \). Unlike square roots, cube roots can be both positive and negative. For example, the cube root of 8 is 2, because \( 2^3 = 8 \). You will often see cube roots denoted as \( \sqrt[3]{x} \).

Understanding cube roots is essential in problems involving cube root operations such as multiplication or simplification. When dealing with cube roots, try to factor numbers under the root sign into smaller parts or their prime factors to see if simplification is possible.

Multiplication of Roots

When multiplying cube roots, remember the property \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b} \). This property simplifies the process by allowing you to multiply the numbers under the root signs first before taking the cube root of the result.

For instance, if you're asked to multiply \( \sqrt[3]{6} \times \sqrt[3]{16} \), you first multiply 6 and 16 to get \( \sqrt[3]{96} \).

This approach not only simplifies the multiplication process but also sets the stage for potential simplification of the radical expression. Make sure to perform multiplication carefully to avoid errors that could lead to incorrect results.

Simplification of Radicals

Simplifying radicals, especially cube roots, involves reducing them to their simplest form. To simplify \( \sqrt[3]{96} \), check to see if 96 can be expressed as a product of perfect cubes and other numbers. While 96 itself is not a perfect cube, you can break it down into prime factors: \( 96 = 2^5 \times 3^1 \).

In some cases, numbers can be rewritten as a product of a perfect cube and another number, allowing further simplification. For higher level simplifications, recognize patterns or small cubes, such as \( 8 = 2^3 \) or \( 27 = 3^3 \). By identifying these, you can simplify the cube root by taking the cube root of the perfect cube part out of the radical.

Ultimately, simplification is about making the expression as concise as possible. If no further simplification is feasible, such as with \( \sqrt[3]{96} \) in this exercise, it's already in its simplest form.