Problem 19 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ (3 \sqrt[3]{9})(2 \sqrt[3]{3}) $$

Step-by-Step Solution

Answer

The expression simplifies to 18.

Step 1: Multiply the Coefficients

Multiply the numerical coefficients outside the radicals. We have 3 and 2, so their product is:\[ 3 imes 2 = 6 \]

Step 2: Multiply the Radicands

Multiply the expressions inside the cube roots. The radicands are 9 and 3, thus:\[ 9 imes 3 = 27 \]

Step 3: Write as a Single Expression

Combine the results from Step 1 and Step 2 into a single expression:\[ 6 imes \sqrt[3]{27} \]

Step 4: Simplify the Cube Root

Simplify the cube root if possible. The cube root of 27 is:\[ \sqrt[3]{27} = 3 \]because \(3^3 = 27\).

Step 5: Final Multiplication

Multiply the coefficient 6 by the simplified cube root 3:\[ 6 imes 3 = 18 \]

Key Concepts

Simplifying Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and higher-order roots. Simplifying radicals involves breaking down the root into smaller parts to make the expression easier to handle.
For cube roots, like in our example, we look for values that can be cubed to result in the expressions under the root sign.
Here's a simple process to follow:

• Identify a perfect cube that is a factor of the radicand. This helps to rewrite the root more simply.
• Extract the cube root of this perfect cube, turning it into a regular coefficient outside the radical.
• If the radicand isn't a perfect cube or doesn't contain one as a factor, then simplifying further isn't possible, and it remains under the root sign.

Understanding how to simplify radicals not only makes equations easier to solve but also helps clarify the relationships between numbers in the expression.

Multiplying Coefficients

In expressions with radicals, each term consists of a coefficient and a radicand. The coefficient is the number in front of the radical, while the radicand is under the root. In the process of multiplying these expressions, the coefficients are multiplied separately from the radicals.
Here's a breakdown of how this works:

• Multiply the coefficients directly. For instance, if you have coefficients 3 and 2, their product is 6.
• Keep this product as a separate factor in your expression. This helps in organizing your steps and ensures you multiply only the corresponding parts together.
• After multiplying the coefficients, you can then proceed to multiply the radicands, as seen in our exercise.

This process helps maintain neatness in your calculations and minimizes the chance of errors.

Cube Roots

The cube root function is the inverse of cubing a number. It asks what number, when multiplied three times by itself, gives the original number under the root.
Cube roots are particularly useful in expressions that involve powers of three.
Working with cube roots has its own set of steps:

• First, determine if the number under the cube root is a perfect cube. This makes simplification straightforward.
• If it's a perfect cube, it can be simplified directly. For example, the cube root of 27 is 3 because 3 multiplied by itself three times is 27.
• If it's not a perfect cube, check if it can be broken into factors, including a perfect cube.

Mastering the handling of cube roots across different expressions builds a great foundation for more advanced algebraic concepts.