Problem 28 - Intermediate Algebra

Question

Find each cube root. $$ \sqrt[3]{-64 x^{6}} $$

Step-by-Step Solution

Answer

The cube root of \(-64x^6\) is \(-2x^2\).

Step 1: Recognize the Cube Root

The cube root of a number or expression, written as \( \sqrt[3]{A} \), is the value that, when multiplied by itself three times, gives \( A \). Here, we are finding the cube root of \(-64x^6\).

Step 2: Break Down the Expression

The expression \(-64x^6\) can be seen as the product of \(-64\) and \(x^6\). We'll find the cube root of each component separately: \(\sqrt[3]{-64}\) and \(\sqrt[3]{x^6}\).

Step 3: Find the Cube Root of -64

\(-64\) is \(-2^3\), so its cube root \(\sqrt[3]{-64} = -2\). This is because \((-2)\times(-2)\times(-2) = -8\).

Step 4: Find the Cube Root of x^6

\(x^6\) is equivalent to \((x^2)^3\). Therefore, the cube root \(\sqrt[3]{x^6} = x^2\), since \((x^2)\times(x^2)\times(x^2) = x^6\).

Step 5: Combine the Results

Now, combine the individual roots: \(\sqrt[3]{-64x^6} = \sqrt[3]{-64} \times \sqrt[3]{x^6} = -2 \times x^2 = -2x^2\).

Key Concepts

Polynomial Expressions

A polynomial expression is a sum of terms, where each term consists of a coefficient multiplied by variables raised to a power. Polynomials can have one or more terms, and they serve as fundamental building blocks in algebra.

In our exercise, the expression we are dealing with is \(-64x^6\). This polynomial consists of just one term with:

• A constant coefficient \(-64\)
• A variable raised to a power \(x^6\)

Identifying polynomial expressions is crucial as it allows us to handle each part of the expression individually, like calculating roots in this case. Understanding how to work with polynomials facilitates the simplification, manipulation, and solving of mathematical expressions in algebra.

Negative Numbers

Negative numbers are less than zero and are usually found on the left side of the number line. They are represented with a negative sign (-).

When dealing with negative numbers in operations such as taking a cube root, it is important to understand:

• The cube root of a negative number results in a negative number. This differs from square roots, where the root is usually imaginary unless complex numbers are used.

For example, when we discover the cube root of \(-64\), we find \(-2\) because \((-2) imes (-2) imes (-2) = -8\). Our understanding of how negative numbers work allows us to solve expressions like \( \sqrt[3]{-64} \) effectively.

Exponent Rules

Exponent rules are a set of guidelines that help us manipulate expressions involving powers of numbers or variables. These rules are essential to simplify expressions and solve equations in algebra efficiently.

Some useful ones include:

• Product of Powers rule: \(a^m imes a^n = a^{m+n}\)
• Power of a Power rule: \((a^m)^n = a^{m imes n}\)
• Power of a Product rule: \((ab)^n = a^n imes b^n\)

In the given exercise, the expression \(x^6\) is simplified using the Power of a Power rule. We rewrite \(x^6\) as \((x^2)^3\). This enables us to find the cube root easily because \(\sqrt[3]{x^6} = x^2\). Grasping these rules is crucial for simplifying and solving polynomial expressions with powers.

Algebraic Simplification

Algebraic simplification entails reducing an expression to its simplest form. This involves using operations like factoring, expanding, and reducing exponents to make an expression as straightforward as possible.

In the original exercise, simplification of the cube root of \(-64x^6\) is achieved by breaking down the expression into simpler parts and addressing each one individually.

• First, we separated the constant \(-64\) and the variable part \(x^6\).
• Then, we found the cube roots: \(\sqrt[3]{-64} = -2\) and \(\sqrt[3]{x^6} = x^2\).
• Finally, we combined these results to achieve the simplified expression: \(-2x^2\).

By following clear steps and utilizing learned rules, one simplifies any expression efficiently, ultimately solving any algebraic problem with ease.