Problem 70 - Algebra A Combined Function

Question

Assume that all variables represent positive real numbers. $$ \sqrt[3]{-8 a^{21} y^{6}} $$

Step-by-Step Solution

Answer

The cube root simplifies to \(-2a^7y^2\).

Step 1: Recognize the Expression

The expression given is \( \sqrt[3]{-8 a^{21} y^{6}} \). This is a cube root operation on a negative expression, which can be simplified by factoring out the cube root of each constituent part.

Step 2: Simplify the Coefficient

First, simplify the coefficient \(-8\). We know that \(-8 = (-2)^3\). Therefore, the cube root of \(-8\) is \(-2\).

Step 3: Simplify the Variables

Now, simplify the variable components one by one:- For \(a^{21}\), we have \( (a^7)^3 = a^{21} \). So, the cube root of \(a^{21}\) is \(a^7\).- For \(y^6\), we have \( (y^2)^3 = y^6 \). Thus, the cube root of \(y^6\) is \(y^2\).

Step 4: Combine Simplified Parts

Combine all parts simplified in previous steps: the coefficient, and the variables. The cube root simplifies to:\[ \sqrt[3]{-8 a^{21} y^{6}} = -2a^7y^2 \].

Key Concepts

Simplifying Expressions

Simplifying expressions is about breaking down a complex mathematical expression into its simplest form where it is easier to work with or understand. This process often involves removing parentheses, combining like terms, and using mathematical operations efficiently. In the given exercise, we are tasked with simplifying the cube root of a complex expression. To achieve simplification, we utilize the properties of cube roots and factored values. When simplifying any radical expression, like cube roots, our main goal is to express it in the most reduced form possible. This involves:

• Identifying parts of the expression that are perfect cubes.
• Decomposing these parts and then simplifying them individually.

In practice, simplifying expressions not only helps in solving math problems faster but also allows clear understanding and further manipulations like solving equations or substitution, which are key in higher-level mathematics. This simplification is crucial for accurately evaluating mathematical statements as seen in the process of solving the exercise.

Exponents

In mathematics, exponents are used to represent repeated multiplication of the same number or variable. They are expressed as the power of a number or a variable. In this exercise, exponents play a crucial role as each variable is raised to a power, i.e., exponent.For example:

• The expression contains negative and positive exponents like \(a^{21}\) and \(y^6\). Understanding exponents allows for simplifying each term effectively.
• Exponents follow specific properties which help in simplification. Namely, \( (x^m)^n = x^{mn} \).

Relating back to our problem, we encounter two variable expressions with exponents: \(a^{21}\) and \(y^6\). Each of these can be rewritten to reveal parts that are perfect cubes making use of the exponent multiplication rule. This characteristic becomes handy as, by rewriting \(a^{21}\) as \((a^7)^3\), and \(y^6\) as \((y^2)^3\), it makes it easy to extract the cube root.

Factoring

Factoring is the process of breaking down an expression into a product of simpler terms or factors. It's a crucial step in simplifying complex algebraic expressions and solving equations. Factoring becomes particularly useful when dealing with polynomial expressions or when trying to simplify radicals, such as cube roots.In this exercise, factoring helps to break down the expression \(-8 a^{21} y^{6}\) into manageable parts. Here's how:

• The number \(-8\) is identified as \((-2)^3\), a product of its basic factors, making it easily simplify into a single factor.
• Exponents in the expression are also addressed by identifying groups that form a perfect cube. For example, \(a^{21} = (a^7)^3\).

By recognizing these factors, we simplify the process of taking cube roots. Factoring not only eases the simplification process but also permits the transformation of complex expressions into forms that are easier to manipulate and understand when solving complex algebraic problems.