Problem 12 - Algebra 2

Question

Simplify. Assume that all variables are positive. $$ \sqrt[3]{32 a^{5}} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{32 a^5}\) is \(2a\sqrt[3]{4a^2}\)

Step 1 - Rewrite 32 as a cube

We can express 32 as \(2^5\) or \(2^3*2^2\). Rewrite the expression: \(\sqrt[3]{2^3*2^2*a^5}\)

Step 2 - Rewrite a^5 as a product of a cube and a square

Express \(a^5\) as \(a^3*a^2\). So the expression becomes: \(\sqrt[3]{2^3*2^2*a^3*a^2}\)

Step 3 - Simplify the expression

Since the cube root of a cube number gives the base number, the cube root of \(2^3\) is 2, and the cube root of \(a^3\) is a. Hence our simplified form is \(2*a*(\sqrt[3]{2^2*a^2})\)

Key Concepts

Cube Root Simplification

When we talk about cube root simplification, we're referring to the process of simplifying expressions under a cube root. A cube root asks the question: "What number, multiplied by itself three times, will give me the original number?" This means finding the base number of perfect cubes. For example, the cube root of 27 is 3 because

• 3 multiplied by itself three times equals 27

Similar principles apply to variables. If you have something like \(a^3\), the cube root simplifies to \(a\). In cube root simplification, the key step is to identify perfect cubes within the expression. Once recognized, extract these cubes out, leaving a simpler form. Simplifying expressions with cube roots often involves factorizing numbers or variables into powers divisible by three.
This step is crucial in order to separate the cube from the non-cube part, ultimately giving you a cleaner and more aesthetic expression.

Expressions with Exponents

In mathematics, expressions with exponents are useful for representing how many times a number, called the base, is multiplied by itself. Exponents are the tiny numbers we write above and to the right of our base numbers. By understanding how exponents work, you can handle more complex expressions smoothly. Here's a quick dive into the basics:

• An exponent of 2 (squared) means the base is used twice in a multiplication
• An exponent of 3 (cubed) means the base is used three times
• Larger exponents indicate repeated multiplication that many times

It's like a shorthand for multiplication, making it both efficient and easy to read, especially with large numbers. In our expression \(32 a^5\), we broke down the base of 32 into \(2^5\). Then, using properties of exponents, such as \(a^m \times a^n = a^{m+n}\), helps to redistribute and simplify within the cube root context. Remember, mastery of exponents allows you to quickly reshape and simplify expressions, combining base strengths by adding or redistributing power values.

Polynomials

Polynomials are essentially expressions made up of adding or subtracting terms based on variables and exponents. Each term includes coefficients, variables raised to whole-number exponents, but no variables in the denominator. A polynomial can be a single number, a single variable, or the product of several terms.
For example, \(3x^2 + 2x + 1\) is a polynomial. Working with polynomials involves understanding how these terms interact, combining like terms, and applying algebraic rules.

• Coefficients are the numbers in front of variables (like the 3 in \(3x^2\))
• Like terms in polynomials have the same variables raised to the same power and can be combined
• Degree of polynomial is defined by the highest exponent

In expressions involving radicals and polynomials, understanding each term's role helps break down, simplify, and rearrange the components efficiently. In our case, even when simplification under the cube root involves variables raised to an exponent, the principles of dealing with polynomials guide us through the process.