Problem 14 - Algebra for College Students

Question

Multiply and simplify where possible. \((9 \sqrt[3]{6})(2 \sqrt[3]{9})\)

Step-by-Step Solution

Answer

The expression simplifies to \(54 \sqrt[3]{2}\).

Step 1: Distribute the Multiplication

Start by multiplying the coefficients (numbers outside the cube roots) and the expressions inside the cube roots separately. Here, multiply the coefficients:\(9 \times 2 = 18\). Next, multiply the expressions inside the roots:\(\sqrt[3]{6} \times \sqrt[3]{9} = \sqrt[3]{6 \times 9}\).

Step 2: Multiply the Radicals

Calculate the product of the numbers inside the cube roots. \(6 \times 9 = 54\).Thus, the expression becomes:\(18 \sqrt[3]{54}\).

Step 3: Simplify the Radical

Now simplify \(\sqrt[3]{54}\) by factoring 54 into its prime factors.54 can be factored as \(2 \times 3^3\).This allows the cube root to be simplified:\(\sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = 3 \sqrt[3]{2}\), as 3 is a perfect cube and comes out of the radical.

Step 4: Calculate the Simplified Expression

Substitute back to find the entire simplified expression. Multiply the integer outside by the cube root simplification:\(18 \times 3 \sqrt[3]{2} = 54 \sqrt[3]{2}\).

Key Concepts

Simplifying Radicals

Simplifying radicals means reducing the expression under a radical sign to its simplest form. When dealing with cube roots, the goal is to express the number inside the radical with its prime factors and move any perfect cubes out of the radical. To begin, factor the number under the cube root into prime numbers. For example, consider the cube root of 54. The prime factorization of 54 is:

• 54 = 2 × 3 × 3 × 3 = 2 × 3^3

Once you have this factorization, look for groups of three of a kind. Here, you find that 3^3 is a perfect cube. Since cube roots look for groups of three, 3^3 can be taken out of the radical as 3. Consequently, the cube root of 54 simplifies to 3 √[3]{2}. This process simplifies the expression under the cube root and makes it easier to work with. The key steps are to:

• Identify the prime factors.
• Extract any perfect cube groups from the radical.
• Simplify the remaining expression.

Multiplying Radicals

Multiplying radicals involves both the coefficients (the numbers outside the radical) and the radicands (the expressions inside the radical). Always treat these elements separately to simplify multiplication.A useful principle is that the product of \(\sqrt[3]{a} \times \sqrt[3]{b}\) equals \sqrt[3]{a \times b}\. Following this rule, first multiply the coefficients, then the numbers inside each radical, and finally combine everything. Take the expression:

• (9 \sqrt[3]{6})(2 \sqrt[3]{9})

Multiply the coefficients:

• 9 \times 2 = 18

Multiply the radicands:

• \sqrt[3]{6} \times \sqrt[3]{9} = \sqrt[3]{54}

Thus, the expression is now 18 \sqrt[3]{54}. Notice that by combining similar radicals, we can streamline complex expressions into a more manageable form.

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. This technique is essential for simplifying radicals and is especially handy when working with cube roots. To factor a number like 54 into its prime components, begin by dividing by the smallest prime number, continuing the process with the quotient until only prime numbers remain. Here's how you would factor 54:

• 54 is divisible by 2, giving 27.
• 27 is divisible by 3 once, resulting in 9.
• 9 is divisible by 3 twice, resulting finally in 3.
• Thus, 54 can be expressed as 2 × 3 × 3 × 3 or 2 × 3^3.

Prime factorization allows us to see which groups of numbers can be extracted from a radical, aiding in the simplification of cube roots or other radicals. The ability to break numbers down into primes is invaluable in simplifying expressions and solving equations that involve radicals.