Problem 20 - Intermediate Algebra

Question

Find each cube root. $$ \sqrt[3]{27} $$

Step-by-Step Solution

Answer

The cube root of 27 is 3.

Step 1: Understanding the Cube Root

A cube root is a number that, when multiplied by itself three times, gives the original number. For example, if you have a number \( x \), then \( x^3 \) should equal the original number. We are asked to find the cube root of 27.

Step 2: Identifying Candidates

To find the cube root of 27, consider what number multiplied by itself three times results in 27. Recall some common cubes: \( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), and so on.

Step 3: Matching the Cubed Values

Since \( 3^3 = 27 \), we see that the number 3, when cubed, gives us 27. Thus, 3 is our candidate for the cube root.

Step 4: Conclusion

After checking the possible candidates, we find that the cube root of 27 is 3, because \( 3 imes 3 imes 3 = 27 \).

Key Concepts

Cube Numbers

Cube numbers are the result of multiplying a number by itself twice more, for a total of three multiplications. If you take any number, say 2, and multiply it by itself twice, you achieve the cube of that number. In mathematical terms, the cube of a number **n** is represented as \( n^3 \). For example:

• \( 1^3 = 1 \times 1 \times 1 = 1 \)
• \( 2^3 = 2 \times 2 \times 2 = 8 \)
• \( 3^3 = 3 \times 3 \times 3 = 27 \)

Cube numbers might seem simple at first glance, but they establish a foundation for understanding more complex mathematical concepts. Recognizing cube numbers quickly can help you solve problems related to volume in geometry, scaling in algebra, and more.
It's beneficial to memorize the cubes of small numbers, as this helps in identifying cube roots efficiently.

Exponents

Exponents are a mathematical way to express repeated multiplication of the same number. When you see a number like \( x^n \), this tells you to multiply **x** by itself **n** times. In this template, **x** is called the base, and **n** is the exponent. For example:

• \( 2^3 \) means \( 2 \times 2 \times 2 = 8 \)
• \( 5^2 \) means \( 5 \times 5 = 25 \)
• \( 10^1 \) means simply 10, since any number raised to the power of 1 is itself

Exponents provide a shorthand notation that simplifies mathematical operations and equations. Understanding exponents is key to grasping concepts such as scientific notation, polynomial functions, and exponential growth.
In the specific context of cube roots, the cube root \( \sqrt[3]{x} \) is asking what number raised to the power of 3 gives us **x**.

Mathematical Operations

Mathematical operations are basic procedures used to perform calculations. The primary operations include addition, subtraction, multiplication, and division. Each of these operations has certain rules and properties that make it unique. Cube roots, like other root operations, involve a combination of these basic operations.
While performing calculations involving roots, multiplication plays a significant role. For example, finding the cube root of a number involves identifying what number multiplied by itself three times equals the original number.
Let's recap the steps often involved in solving problems using mathematical operations:

• **Identify the operation**: Determine if the problem calls for addition, subtraction, multiplication, or division—or a root operation like cube roots.
• **Apply properties**: Use knowledge of properties of numbers, such as associative and distributive properties, to aid in calculation.
• **Execute the operation**: Carry out the multiplication, division, or finding of a root as necessary.

Understanding these operations makes dealing with cube numbers and exponents more intuitive, allowing for efficient problem-solving. This is especially helpful in breaking down complex equations into manageable steps.