Problem 8 - Algebra 2

Question

Multiply, if possible. Then simplify. $$ \sqrt[3]{-12} \cdot \sqrt[3]{-18} $$

Step-by-Step Solution

Answer

The result of \(\sqrt[3]{-12} \cdot \sqrt[3]{-18}\) is approximately \(5.997\).

Step 1: Find the cube roots

The cube root of a number, denoted as \(\sqrt[3]{x}\), is a value that, when cubed (raised to the power 3), gives the number. So, first find the cube roots of \(-12\) and \(-18\) individually. Now, \(\sqrt[3]{-12}\) is \(-2.289\) (rounded to 3 decimal places) and \(\sqrt[3]{-18}\) is \(-2.620\).

Step 2: Multiply the cube roots

Now we multiply the two cube roots, i.e., we perform \(-2.289 \times -2.620\). When a negative number is multiplied by another negative number, the product turns out to be positive.

Step 3: Simplify the result

After performing the multiplication, we get a result of \(5.997\) (rounded to 3 decimal places). The result cannot be simplified further.

Key Concepts

Understanding Radicals

Radicals are an essential part of mathematics and are particularly helpful for simplifying expressions and solving equations. A radical expression involves roots, such as square roots, cube roots, and other higher roots. An expression with a radical symbol \( \sqrt{} \) is called a radical expression. In this particular exercise, we specifically deal with cube roots, as shown by the symbol \( \sqrt[3]{} \). Cube roots ask the question, "What number, when multiplied by itself three times, gives the original number?"
For numbers like \(-12\) and \(-18\), we indicate their cube roots using \( \sqrt[3]{-12} \) and \( \sqrt[3]{-18} \). Negative numbers can have real cube roots, as the multiplication of three negative numbers results in a negative number. Learning how to work with radicals, especially cube roots in both positive and negative contexts, opens the door to understanding more complex mathematical concepts.

Multiplication of Radical Expressions

When you have two radical expressions, such as \( \sqrt[3]{-12} \cdot \sqrt[3]{-18} \), you can multiply them following a straightforward method. This operation is similar to multiplying any two algebraic expressions. In essence, you multiply the two radicals directly.
In a cube root multiplication scenario, like our exercise, you find each cube root individually, multiply the resulting values, and consider the rules for signs. For negative root values, remember that multiplying two negative numbers gives a positive product. This is why, in our example, the multiplication of \( -2.289 \) and \( -2.620 \) results in a positive value \( 5.997 \).
Understanding how multiplication affects radicals allows students to handle complex algebraic expressions and often simplifies tasks such as equation solving or function evaluation.

Simplifying Radical Expressions

Simplifying a radical expression typically involves reducing it to its simplest form. This could mean finding an exact answer, removing unnecessary parts, or simplifying numerical values. With cube roots, simplification might not always bring an integer or a nice fraction, but it still helps clarify the expression.
In our exercise, after multiplying \( \sqrt[3]{-12} \cdot \sqrt[3]{-18} \) and calculating the product as \( 5.997 \), there isn’t much left for additional simplification in terms of numbers, unless more precise measurements are required. This result is already simplified since it directly expresses the multiplication of the cube roots of the original values.
Simplifying radical expressions enhances the clarity and efficiency of mathematical solutions, crucial for both pure mathematics and practical applications. Ensuring radicals are simplified helps in comparing solutions or further computations with ease.