What is the simplified form of 3 startroot 135 endroot – Embarking on an exploration of “What is the Simplified Form of 3√135”, this discourse delves into the realm of cube roots, their properties, and their applications. Understanding the concept of square roots serves as a cornerstone for grasping the intricacies of cube roots, paving the way for a comprehensive exploration of this mathematical operation.
Delving deeper into the topic, we will unravel the process of simplifying cube roots, shedding light on the nuances of perfect cubes and their role in this mathematical endeavor. Additionally, we will illuminate the steps involved in simplifying expressions containing cube roots, equipping readers with the tools to navigate these mathematical expressions with confidence.
What is the Simplified Form of 3√135?: What Is The Simplified Form Of 3 Startroot 135 Endroot
Understanding the concept of square roots and cube roots is essential for simplifying expressions involving these operations. This article provides a comprehensive guide to simplifying cube roots, including the mathematical definitions, steps, and applications.
Understanding the Concept of Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, since 3 x 3 = 9. Square roots can be represented using the radical symbol √, such as √9 = 3.
Simplifying Cube Roots
A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, since 3 x 3 x 3 = 27. Cube roots are represented using the radical symbol ∛, such as ∛27 = 3.
Simplifying Expressions with Cube Roots
To simplify expressions involving cube roots, follow these steps:
- Factor the radicand into prime factors.
- Group the factors into sets of three, starting with the perfect cubes.
- Take the cube root of each perfect cube factor and multiply the results together.
- Leave any remaining factors as they are.
Examples and Applications, What is the simplified form of 3 startroot 135 endroot
Here are some examples of simplifying cube roots:
| Expression | Simplified Form |
|---|---|
| ∛125 | 5 |
| ∛216 | 6 |
| ∛54 | 3∛2 |
Cube roots have applications in various fields, such as geometry and volume calculations. For example, the volume of a cube is calculated using the formula V = a³, where a is the length of one side of the cube. To find the length of the side of a cube with a volume of 216 cubic units, you would take the cube root of 216, which is 6.
General Inquiries
Q: How do I simplify a cube root expression?
A: To simplify a cube root expression, factor the radicand into its prime factors and group them into cube factors. Then, extract the cube root of each cube factor and multiply the results.
Q: What is the simplified form of 3√135?
A: The simplified form of 3√135 is 3√(27 × 5) = 3 × 3√5 = 9√5.
Q: How are cube roots used in real-world applications?
A: Cube roots find applications in various fields, including geometry (calculating volumes of cubes and spheres) and physics (determining the acceleration of objects in free fall).
