Problem 36 - Algebra 2

Question

Simplify. $$ \sqrt[3]{-27 c^{9} d^{12}} $$

Step-by-Step Solution

Answer

The simplified form is \\(-3c^3d^4\\).

Step 1: Understand the Cube Root

The expression involves finding the cube root of \(-27c^9d^{12}\). Recall that the cube root of a number \(a^3\) is equal to \(a\). We can apply this property to simplify each component under the cube root.

Step 2: Simplify the Cube Root of -27

The number \(-27\) can be rewritten as \((-3)^3\) because \((-3) imes (-3) imes (-3) = -27\). Thus, the cube root of \(-27\) is \(-3\).

Step 3: Simplify the Cube Root of c^9

The exponent rule states that \(\sqrt[3]{c^9}\) simplifies to \(c^{9/3} = c^3\) by dividing the exponent by 3.

Step 4: Simplify the Cube Root of d^{12}

Similarly, \(\sqrt[3]{d^{12}}\) simplifies to \(d^{12/3} = d^4\) by dividing the exponent by 3.

Step 5: Combine the Results

Now combine all the simplified results: The cube root of \(-27c^9d^{12}\) is equal to \(-3c^3d^4\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them into their simplest forms. In this task, we are working with the cube root of an expression. The goal is to find equivalent terms that make the expression easier to handle.

• These equivalent terms usually use the simplest coefficients and the most reduced form of each variable.
• This can include combining like terms and simplifying exponents using rules.

Begin by breaking down the original expression. For example, when simplifying \( \sqrt[3]{-27c^9d^{12}} \), tackle each component separately. Understanding how numbers and variables behave differently under root operations is crucial. Each part of the expression will be simplified independently before putting it back together.Next, recognize parts of the expression that are perfect cubes, such as \(-27\). This aids in simplifying its cube root immediately. Similarly, use the exponent rules for terms like \(c^9\) and \(d^{12}\) to simplify them effectively.Breaking down expressions keeps you organized. It ensures clarity at each step, helping avoid mistakes and leading to the simplest version of your mathematical expression.

Exponent Rules

Exponent rules are helpful when simplifying expressions involving powers. They define how to handle powers in different mathematical scenarios, and they are key to solving expressions like \( \sqrt[3]{-27c^9d^{12}} \).### Basic Exponent Rules

• The **power of a power rule** says when raising a power to another power, multiply the exponents. For example, \((a^m)^n = a^{mn}\).
• The **product of powers rule** indicates to add exponents when multiplying like bases: \(a^m \cdot a^n = a^{m+n}\).
• The **quotient of powers rule** tells us to subtract exponents when dividing like bases: \(\frac{a^m}{a^n} = a^{m-n}\).

In our solution, we used the rule for taking roots: when you take a root of a power, you divide the exponent by the root's index. So, \(\sqrt[3]{c^9}\) becomes \(c^{9/3}\), which simplifies to \(c^3\). Similarly, \(\sqrt[3]{d^{12}}\) turns into \(d^{12/3} = d^4\).Knowing these rules helps break down complex expressions, so you can solve them systematically. Mastering exponent rules simplifies tackling much bigger mathematical problems.

Negative Numbers

Understanding negative numbers is important for expressions, especially when taking roots. When simplifying \( \sqrt[3]{-27c^9d^{12}} \), we encounter a negative number.### Working with Negative Numbers

• Negative numbers are numbers less than zero, and they have a negative sign, \(-\), in front of them.
• Cube roots of negative numbers remain negative. This differs from square roots where negative results are not possible under real numbers.

For instance, the cube root of \(-27\) is \(-3\). This is because multiplying \(-3\times -3\times -3 = -27\). Cubes preserve the sign of the original number because it's odd; hence odd roots maintain negative values.Understanding how negative numbers operate, especially in root operations, eliminates common pitfalls — such as incorrectly changing signs. When solving equations or expressions, this clarity around negatives ensures your answers are both mathematically accurate and as intended.