Problem 28 - Algebra 2

Question

Simplify. $$ \sqrt[3]{-c^{6}} $$

Step-by-Step Solution

Answer

\((-c^{2})\)

Step 1: Understand the Cube Root

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. For cube roots, the symbol is \( \sqrt[3]{...} \).

Step 2: Simplify the Expression Inside the Cube Root

The expression inside the cube root is \( -c^{6} \). We can rewrite this as \( (-1) \cdot (c^{6}) \).

Step 3: Take the Cube Root of -1

The cube root of \( -1 \) is \( -1 \) because \( (-1) \cdot (-1) \cdot (-1) = -1 \). So, \( \sqrt[3]{-1} = -1 \).

Step 4: Simplify the Cube Root of \\( c^{6} \\\\)

Taking the cube root of \( c^{6} \) gives us \( c^{6/3} = c^{2} \). This is because \( c^{6} = (c^{2})^{3} \).

Step 5: Combine the Results

Combine the results from Steps 3 and 4: \( \sqrt[3]{-c^{6}} = (-1) \cdot c^{2} = -c^{2} \).

Key Concepts

Cube Roots

Understanding cube roots is essential to simplifying expressions like \( \sqrt[3]{-c^6} \). A cube root helps us find a number that, when multiplied by itself twice, gives us the original value. For example, if we have \( x = \sqrt[3]{y} \), then \( x \cdot x \cdot x = y \). Cube roots are represented using the symbol \( \sqrt[3]{...} \). Here, the '3' indicates that we are trying to find a cube root, as opposed to a square root which would be \( \sqrt{...} \).

• Cubes involve multiplying a number three times by itself.
• Cube roots help reverse this operation, revealing the original base number.

In our example, \( \sqrt[3]{-c^{6}} \), we applied cube roots to both \(-1\) and \(c^6\). The expression inside the cube root is decomposed and simplified individually, making it easier to work with complex numbers.

Exponents

Exponents are a way to express repeated multiplication of the same number. For instance, \(a^6\) means \(a \cdot a \cdot a \cdot a \cdot a \cdot a\). They are crucial in scientific notation and algebra for simplifying complex expressions. When dealing with cube roots like in our example \(\sqrt[3]{-c^6}\), it's important to understand how exponents interact with roots.

• Using exponents enables us to express large numbers more easily.
• They are tightly connected with roots, helping us simplify roots like cube roots and square roots.

In the expression \(c^6\), we used the relationships between exponents and cube roots. To solve \(\sqrt[3]{c^6}\), convert the exponents into fractional form: \(c^{6/3} = c^2\). Here, the cube root extracts a third root, effectively simplifying the expression.

Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign \(-\) in front. Understanding them is important in algebra and complex calculations. In the context of cube roots, negative numbers have interesting properties. Let's explore.

• Negative numbers can be multiplied three times to give a negative result, unique to cube roots.
• They often appear in real-life calculations, adding depth to graph plots and real processes.

In our solved exercise with \(\sqrt[3]{-1}\), this property of negative numbers helps us directly. The cube root of a negative number like \(-1\) is still \'-1\' because \((-1) \cdot (-1) \cdot (-1)\) matches \(-1\). This is a critical feature allowing us to work smoothly with negatives within cube root expressions.