Problem 50 - College Algebra with Modeling and Visualization

Question

Simplify the radical expression by factoring out the largest perfect nth power. Assume that all variables are positive. $$ \sqrt[3]{-81 a^{5} b^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-3a\sqrt[3]{3a^2b^2}.\)

Step 1: Factor the Expression inside the Radical

Start by factoring the expression inside the cube root. Identify the prime factorization of -81, which is \(-81 = -1 \times 3^4\). Therefore, \(-81 a^5 b^2 = -1 \times 3^4 \times a^5 \times b^2.\)

Step 2: Group the Perfect Cube Factors

Next, group the factors into perfect cubes. Remember, a perfect cube has the form where the exponent is a multiple of 3. Here we have \(-1 = (-1)^1,\)\(3^4 = 3^3 \times 3,\)\(a^5 = a^3 \times a^2,\)\(b^2 = b^2\).

Step 3: Simplify the Cube Root

Simplify \(\sqrt[3]{-1}\) as \(-1,\)\(\sqrt[3]{3^3} = 3.\)Thus, \(\sqrt[3]{-81 a^5 b^2} = (-1) \times 3 \times \sqrt[3]{3 \times a^2 \times a^3 \times b^2}.\)

Step 4: Final Simplification

Extract the remaining perfect cubes: \(\sqrt[3]{a^3} = a.\)Thus, the expression simplifies to:\(-3a\sqrt[3]{3a^2b^2}.\)

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This process helps to simplify complex expressions and identify components that can be further broken down, especially in expressions involving radicals.

To find the prime factorization, you repeatedly divide the number by its smallest prime factor until you're left with 1. For example, to factor -81:

• Start with the smallest prime number, which is 3, and check divisibility: 81 can be divided by 3.
• Divide 81 by 3, which gives 27. Divide 27 again by 3 to get 9, then by 3 to get 3, and finally 3 by 3 to get 1.
• Thus, the prime factorization of 81 is 3 raised to the power of 4, because you multiplied 3 by itself four times.
• Don't forget to include the negative sign as -1, making it -1 × 3⁴.

Perfect Cubes

A perfect cube is a number that can be expressed as another whole number raised to the power of three. These values have exponents that are multiples of 3, and they are essential in simplifying cube roots.

When simplifying radical expressions, recognizing perfect cubes allows you to rearrange and group parts of an expression easily. Consider the components like

• -1 which is cube of -1 (since \((-1)^3 = -1\)),
• a misleading part \(3^4\), which can be broken down into \(3^3 imes 3\),
• a^5 which can be rewritten as \(a^3 imes a^2\) given that \(a^3\) is a perfect cube, and
• b² remains as it is since it does not form a perfect cube.

Identifying these perfect cubes simplifies the process of finding cube roots.

Cube Root Simplification

Simplifying cube roots involves extracting factors that are perfect cubes from inside the radical. The cube root of a number asks what number multiplied by itself three times yields the original number.

For the radical expression \(\sqrt[3]{-81a^{5}b^{2}}\):

• Simplify \(\sqrt[3]{-1}\) as \(-1\).
• Recognize \(\sqrt[3]{3^3}\) as 3.
• \(a^3\) gives \(a\) since it’s a perfect cube itself.

After extracting these, the remaining expression is \(\sqrt[3]{3a^2b^2}\). The entire simplified expression becomes \(-3a\sqrt[3]{3a^2b^2}\.\)

Factoring

Factoring is breaking down an expression into a product of simple parts. In the context of simplifying radical expressions, it involves grouping like terms and identifying common factors or perfect powers like squares or cubes.

The key is to first factor the expression as much as possible. In this example, you start with \(-81a^5b^2\), converting primes and rewriting with factors that are perfect powers:

• Write \(-1\) as \((-1)^1\) and \(-81\) as \(-1 \times 3^4\).
• Rearrange terms like \(3^4\) to \(3^3 \times 3\), \(a^5\) as \(a^3 \times a^2\).

This methodical approach ensures clarity in each step, making the complex problem more manageable and revealing the underlying structure of mathematical expressions.