Problem 49 - Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (-2 \sqrt[3]{2 x^{2}})^{3} $$

Step-by-Step Solution

Answer

The result is \(-16x^{2}\).

Step 1: Identify the operation

In this exercise, we need to cube the entire expression, which is \( (-2 \sqrt[3]{2 x^{2}})^{3} \). This means multiplying it by itself three times.

Step 2: Distribute the cube

Recognize the expression as \( a^{3} \) where \( a = -2 \sqrt[3]{2 x^{2}} \). When cubing, use the property \((a ext{b})^{3} = a^{3} ext{b}^{3}\). So, we distribute the cube over both parts: \[ (-2)^{3} (\sqrt[3]{2 x^{2}})^{3} \].

Step 3: Cube each part separately

Calculate each part separately:1. For \((-2)^{3}\), calculate as \((-2) \times (-2) \times (-2) = -8\).2. For \((\sqrt[3]{2 x^{2}})^{3}\), use the cube root property: the cube of the cube root is the original value, so we have \( (\sqrt[3]{2 x^{2}})^{3} = 2x^{2} \).

Step 4: Multiply results

Multiply the results from Step 3: \(( -8) \times (2x^{2}) = -16x^{2}\). This is the simplified result.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation involving numbers, known as the base and the exponent. The base is the number that gets multiplied, while the exponent tells us how many times the base is used as a factor. For example, in the expression \((-2)^3\), \(-2\) is the base and \(3\) is the exponent. This means \(-2\) is multiplied by itself three times: \((-2) \times (-2) \times (-2) = -8\). Exponentiation is a powerful tool in algebra that helps us work with repeated multiplication in a compact form.

Multiplying Expressions

Multiplying expressions is a key part of algebra, involving the application of the distributive property, just like in this problem. When multiplying two expressions, each term in the first expression is multiplied by each term in the second expression. Consider the problem \((-2 \sqrt[3]{2 x^{2}})^{3}\). We distribute the exponent to every factor: \[(-2)^{3} \times (\sqrt[3]{2 x^{2}})^{3}\]. By doing so, we simplify our calculations to two main tasks: finding \((-2)^3\) and \((\sqrt[3]{2 x^{2}})^{3}\). These separate calculations are what we combine to arrive at our final result.

Cube Root

A cube root of a number \(y\) is a number \(x\) such that \(x^3 = y\). It's the number that produces \(y\) when cubed. In this exercise, there's the term \(\sqrt[3]{2 x^2}\). When cubing the cube root of a number, it cancels out the cube root, effectively restoring the original value inside. So, \((\sqrt[3]{2 x^{2}})^{3}\) becomes simply \(2x^{2}\). This simplification is thanks to the inverse relationship between cubing and taking the cube root.

Simplification

Simplification in math involves altering an expression into a simpler or more understandable form without changing its value. It's crucial in algebra to make expressions easier to work with and to derive meaningful insights. In our problem, after calculating \((-2)^{3} = -8\) and \((\sqrt[3]{2 x^{2}})^{3} = 2x^{2}\), we multiply these results to get \(-16x^{2}\). This represents taking our expression through various stages of simplification to reach an answer that is clear and manageable to work with. By consistently applying algebraic rules and properties, we achieve this streamlined result.