Problem 110 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[3]{-216 z^{9}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-6z^3\).

Step 1: Identify Radicand Components

The expression is the cube root of \(-216z^9\). The negative sign can be separated as \(-(216\cdot z^9)\). We need to simplify \(-216\) and \(z^9\) separately.

Step 2: Simplify the Numerical Part

\(-216\) can be expressed as \(-1\times6^3\). Thus, the cube root of \(-216\) is \(-6\) because \(-1^{1/3} = -1\) and \((6^3)^{1/3} = 6\).

Step 3: Simplify the Variable Part

The exponent of \(z\) is 9, which is a multiple of 3. Apply the rule \((z^9)^{1/3} = z^{9/3} = z^3\).

Step 4: Combine the Results

By combining, we can write the simplified expression of \(\sqrt[3]{-216z^9}\) as \(-6\times z^3 = -6z^3\).

Key Concepts

Cube Root Simplification

Understanding how to simplify cube roots can make radical expressions less intimidating. A cube root, denoted as \(\sqrt[3]{a}\), is asking what number, when multiplied by itself three times, gives you \(a\). In our example, we had \(\sqrt[3]{-216}\).

To solve this, we needed to break down \(-216\) into parts where we can identify cubes. Here, \(-216 = -1 \times 6^3\). The cube root of \(-216\) then simplifies to \(-6\), because:

• \((-1)^{1/3} = -1\)
• \((6^3)^{1/3} = 6\)

Cube roots allow negative results because a negative number cubed is negative. This differs from square roots, which don't have real solutions for negatives. Cube root simplification is about recognizing perfect cubes for efficiency.

Variables in Radicals

Variables inside radical expressions follow the same rules as numbers but often seem trickier. When you have a variable like \(z^9\) under a cube root, you're essentially solving \((z^9)^{1/3}\).

Break down the process by looking for exponents that are multiples of the root number. For a cube root, you're looking for multiples of 3.

• For \((z^9)^{1/3}\), recognize that 9 is a multiple of 3.
• This allows us to simplify \((z^9)^{1/3}\) directly to \(z^{9/3} = z^3\).

This method lets you simplify the variable part cleanly and accurately. Always ensure your variable's exponent is correctly interpreted with its root to simplify properly.

Exponent Rules

Exponents are crucial when dealing with radicals, especially for simplification. The radical expression \(\sqrt[3]{z^9}\) uses exponent rules to transform and simplify.

Remember these key points:

• The power rule: \((a^m)^n = a^{m \times n}\), which helps in breaking down expressions under a radical.
• Root conversion: The cube root \(\sqrt[3]{a}\) is the same as \(a^{1/3}\).
• For \(z^9\) in a cube root, transform this to \((z^9)^{1/3} = z^{9 \times (1/3)} = z^3\).

Proper use of exponent rules allows you to manipulate and simplify expressions interchangeably between radical and exponential form, making calculations straightforward. Understanding these rules broadens your mathematical toolkit, especially in algebra.