Problem 12 - Intermediate Algebra

Question

For Problems \(1-14\), multiply and simplify where possible. $$ (4 \sqrt[3]{3})(5 \sqrt[3]{9}) $$

Step-by-Step Solution

Answer

The simplified result is 60.

Step 1: Multiply the Coefficients

Multiply the numbers outside the cube roots, which are 4 and 5. This gives us:\[ 4 \times 5 = 20 \].

Step 2: Use the Product Property of Radicals

According to properties of radicals, \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b} \). Apply this to the cube roots. Multiply the numbers inside the cube roots:\[ \sqrt[3]{3} \times \sqrt[3]{9} = \sqrt[3]{3 \times 9} = \sqrt[3]{27} \].

Step 3: Simplify the Expression

Now simplify \( \sqrt[3]{27} \). Since 27 is a perfect cube (\(3^3\)),\[ \sqrt[3]{27} = 3 \].

Step 4: Final Multiplication

Multiply the simplified cube root by the product of the coefficients:\[ 20 \times 3 = 60 \].

Key Concepts

Understanding the Product Property of Radicals

When dealing with radicals, especially cube roots, the product property can be incredibly helpful. This property states that the product of two radicals with the same index can be combined into one radical.
In simpler terms, if you have \( \sqrt[n]{a} \times \sqrt[n]{b} \), you can rewrite them as \( \sqrt[n]{a \times b} \). This approach simplifies the multiplication process, as you only secure one radical expression.
Let's see this in an example: in the given problem, \( (4 \sqrt[3]{3})(5 \sqrt[3]{9}) \), you multiply the cube roots using this property. So, \( \sqrt[3]{3} \times \sqrt[3]{9} = \sqrt[3]{3 \times 9} = \sqrt[3]{27} \). You switch from two cube root terms to a single one.
By applying the product property, multiplication becomes straightforward and often leads to simplification.

Exploring Cube Roots

Cube roots are intriguing because they break down numbers into a factor that, when multiplied by itself twice more, results in the original number. Mathematically, \( \sqrt[3]{x} \) equals a number that, when cubed, gives back \( x \).
Finding cube roots is like solving a little puzzle backward.

• For example, \( \sqrt[3]{27} \) asks "What number multiplied by itself twice results in 27?" That's 3, since \( 3^3 = 27 \).
• Cube roots can simplify expressions massively if one recognizes a perfect cube.

In the solution process above, simplifying \( \sqrt[3]{27} \) to 3 is a crucial step. Recognizing that 27 forms a perfect cube aids in completing further simplifications.

Simplifying Radicals

Simplifying radicals is all about making expressions simpler and more manageable through calculations. This involves reducing radical numbers or expressions to their simplest form.
Here are tips to simplify:

• First, factor out the expression inside the radical to find any perfect powers that match the radical's index.
• For numbers under cube roots, check if they can be written as a smaller cube. This helps transform the radical into a simple number.

In our problem, you see this clearly with \( \sqrt[3]{27} \).
Since 27 is expressed as a cube of 3 (i.e., \( 3^3 \)), it easily simplifies to 3. This simplification significantly works well when solving the original equation involving multiplication. The process makes complicated expressions much easier to handle, leading to quick solutions.