Problem 79 - Intermediate Algebra

Question

Simplify each cube root. See Example 6. $$ \sqrt[3]{-216 a^{3}} $$

Step-by-Step Solution

Answer

The simplified cube root of \(\sqrt[3]{-216 a^{3}}\) is \(-6a\).

Step 1: Identify the Cube Root Components

We need to simplify the expression \(\sqrt[3]{-216 a^3} \). Recall that cube roots can be simplified by identifying perfect cubes within the expression. Here, \(-216\) is a number and \(a^3\) is already in the form of a perfect cube.

Step 2: Simplify the Numeric Cube Root

Determine if \(-216\) is a perfect cube. The perfect cube closest to \(-216\) is formed by \(-6\), because \((-6)^3 = -216\). Thus, \(\sqrt[3]{-216} = -6\).

Step 3: Simplify the Variable Component

For the variable part, \(a^3\), the cube root is \(a\) since \( (a^3)^{1/3} = a \). The cube root of any expression raised to the power of 3 is simply the base of that expression.

Step 4: Combine and Simplify

Now combine both simplified parts: \(-6\) from the numeric part and \(a\) from the variable part. The simplified expression is thus \(-6a\).

Key Concepts

Perfect Cubes

Understanding perfect cubes is extremely useful in simplifying cube roots. A perfect cube results from multiplying an integer by itself twice: in other words, raising a number to the power of three. For example, the number 8 is a perfect cube because it's equal to \[ 2^3 = 2 \times 2 \times 2 = 8. \] Similarly, \[ 64 \] is a perfect cube because \[ 4^3 = 4 \times 4 \times 4 = 64. \]Further examples include -27 as \[ (-3)^3 \] and 125 as \[ 5^3 \].
When dealing with negative numbers, it's important to remember that these, too, can be perfect cubes if you multiply three identical integers where one or all are negative. For instance, \[ (-6)^3 = -216 \].This means you can simplify \[ \sqrt[3]{-216} \] to \[ -6 \].Thus, identifying perfect cubes can significantly simplify algebraic computations involving cube roots.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest, most readable form. When it comes to cube roots, this often involves recognizing and extracting perfect cubes. To simplify expressions like \( \sqrt[3]{-216 a^3} \),you first simplify the numerical part and then the variable part.
Here's how:

• Identify perfect cubes within the expression. For example, notice that -216 is a perfect cube of -6.
• Calculate the cube root of the numerical perfect cube separately from the variable part.
• For variables with exponents, apply the rule that \( (x^n)^{1/3} = x^{n/3} \),and simplify any terms with exponents that are multiples of 3.

In this specific case, for \( a^3 \),the exponent of 3 can be simplified to \( a \) because \( (a^3)^{1/3} = a^{3/3} = a \).Thus, simplifying expressions requires recognizing patterns and utilizing the rules of exponents efficiently to condense the expression neatly.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations like addition, subtraction, multiplication, and division. When working with cube roots in algebra, such as \( \sqrt[3]{-216 a^3} \),it's important to apply the correct principles to simplify the expression. Here's what to consider:

• Identify and separate the numeric components from the variable components.
• Simplify each part according to its properties: where numeric components become integers, and variable components simplify based on their exponents.
• Recombine the two parts into one simple expression.

In the original problem \( \sqrt[3]{-216 a^3} \),we start by knowing that -216 is a perfect cube of -6. So the numeric part simplifies to -6. For the variable \( a^3 \), the power is a multiple of 3, which simplifies to just \( a \) when we take the cube root. Together, they form the simplified algebraic expression \( -6a \).This combination represents a clear example of working with both numeric and algebraic components within an expression.