Problem 57 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt[3]{125 r^{9} s^{12}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 5r^3s^4 \).

Step 1: Identify the Cube Inside the Radical

The expression given is \( \sqrt[3]{125 r^{9} s^{12}} \). First, observe that the cube root is taken over a product of terms. We know that a cube root can be individually applied to each factor in the product inside the radical.

Step 2: Simplify the Numerical Part

Identify if the number is a perfect cube. The number 125 is a perfect cube because \( 125 = 5^3 \). Hence, \( \sqrt[3]{125} = 5 \). This simplifies the numerical part of the expression.

Step 3: Simplify the Variable Part with Exponents

Apply the cube root to each variable term separately. Use the property \( \sqrt[3]{x^n} = x^{n/3} \). For \( r^9 \), the cube root is \( r^{9/3} = r^3 \). For \( s^{12} \), the cube root is \( s^{12/3} = s^4 \).

Step 4: Combine the Simplified Terms

Combine all the simplified terms together. We have \( 5 \) from the numerical simplification, \( r^3 \) from the \( r^9 \) term, and \( s^4 \) from the \( s^{12} \) term. Therefore, the fully simplified expression is \( 5r^3s^4 \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves turning complex root expressions into simpler, more manageable forms. When we talk about radicals, we refer to expressions that contain a root, such as square roots or cube roots. For example, in the expression \( \sqrt[3]{125 r^{9} s^{12}} \), the radical sign \( \sqrt[3]{} \) denotes the cube root.

To simplify a radical, we can break it down into separate components, just as we did in the original exercise. For the given expression:

• Identify the components inside the radical — this often involves numbers and variables.
• Apply the root to each part, simplifying as much as possible.

The goal is to simplify each component separately and then combine them. Through this process, radicals become easier to understand and work with, allowing for simpler and cleaner mathematical expressions.

Exponents

Exponents play a crucial role in simplifying expressions, especially with radicals. An exponent indicates how many times a number, known as the base, is multiplied by itself. In cube roots, we often encounter expressions like \( x^n \), where \( n \) is the exponent.

When you are taking the cube root of an expression with exponents like \( r^9 \) or \( s^{12} \), you apply the rule: \( \sqrt[3]{x^n} = x^{n/3} \). This means dividing the exponent by 3. As observed:

• \( r^9 \) becomes \( r^{9/3} = r^3 \).
• \( s^{12} \) becomes \( s^{12/3} = s^4 \).

Understanding exponents is key to success in simplifying radicals, as it allows us to break down complex powers into more manageable forms.

Perfect Cubes

Identifying perfect cubes is a fundamental step in simplifying cube roots. A number is considered a perfect cube if it can be written as \( x^3 \), where \( x \) is an integer. In this context:

• For the number 125, we recognize it as a perfect cube because \( 125 = 5^3 \). Thus, the cube root \( \sqrt[3]{125} = 5 \).

Recognizing perfect cubes simplifies the process significantly. It helps to immediately simplify the numerical part of a radical expression without complicating calculations.

In summary, knowing common perfect cubes such as \( 1, 8, 27, 64, 125, \) and so on, can speed up simplification. Once you identify them, you only need to find the number that was cubed, making it straightforward to compute the cube root.