What Is 2 5/3 Raised To The Third Power

What is 2 5/3 Raised to the Third Power? A Comprehensive Guide

This article delves deep into the mathematical problem of raising the mixed number 2 5/3 to the third power (or cubing it). We'll break down the process step-by-step, explore different approaches, and discuss the underlying mathematical concepts involved. This guide aims to be comprehensive, addressing various levels of mathematical understanding.

Understanding Mixed Numbers and Exponents

Before tackling the problem, let's clarify the terms involved.

Mixed Numbers: A mixed number combines a whole number and a fraction, like 2 5/3. This represents 2 + 5/3.

Exponents (or Powers): An exponent indicates how many times a number (the base) is multiplied by itself. For example, 2³ (2 raised to the power of 3 or 2 cubed) means 2 x 2 x 2 = 8.

Our problem: (2 5/3)³ means we need to multiply 2 5/3 by itself three times: (2 5/3) x (2 5/3) x (2 5/3).

Method 1: Converting to an Improper Fraction

The most straightforward method is to convert the mixed number into an improper fraction before calculating the exponent.

Step 1: Convert the Mixed Number to an Improper Fraction

To convert 2 5/3 to an improper fraction, we follow these steps:

1. Multiply the whole number by the denominator: 2 x 3 = 6
2. Add the numerator: 6 + 5 = 11
3. Keep the same denominator: 3

Therefore, 2 5/3 is equivalent to the improper fraction 11/3.

Step 2: Cube the Improper Fraction

Now we cube the improper fraction: (11/3)³ = (11/3) x (11/3) x (11/3)

Step 3: Multiply the Numerators and Denominators

Multiply the numerators together: 11 x 11 x 11 = 1331 Multiply the denominators together: 3 x 3 x 3 = 27

This gives us the improper fraction 1331/27.

Step 4: Convert Back to a Mixed Number (Optional)

While the answer as an improper fraction (1331/27) is perfectly acceptable, we can convert it back to a mixed number for easier interpretation.

1. Divide the numerator by the denominator: 1331 ÷ 27 = 49 with a remainder of 8.
2. The quotient becomes the whole number: 49
3. The remainder becomes the numerator: 8
4. The denominator remains the same: 27

Therefore, 1331/27 is equivalent to the mixed number 49 8/27.

Method 2: Expanding and Multiplying

This method involves expanding the expression and performing the multiplication directly. While potentially more time-consuming, it can enhance understanding of the underlying process.

(2 5/3)³ = (2 5/3) x (2 5/3) x (2 5/3)

We can rewrite 2 5/3 as (6/3 + 5/3) = 11/3.

So, the equation becomes: (11/3) x (11/3) x (11/3)

This directly leads to 1331/27, which, as calculated before, simplifies to 49 8/27.

Understanding the Significance of the Result

The result, 49 8/27, represents the volume of a cube with sides of length 2 5/3 units. This illustrates the practical application of exponents in geometry and other fields. Understanding how to manipulate and simplify these expressions is crucial in various mathematical and scientific disciplines.

Advanced Considerations: Working with Other Exponents

The methods described above can be adapted to handle other exponents. For example, raising 2 5/3 to the power of 4 would involve multiplying 11/3 by itself four times. The principle remains the same: convert to an improper fraction, perform the exponentiation, and then convert back to a mixed number if desired.

Furthermore, these principles extend to other types of numbers. The process of converting to a simpler form (like an improper fraction), performing the calculation, and then converting back to a more intuitive form is applicable to numerous mathematical problems involving exponents and various number types.

Practical Applications and Real-World Examples

The concept of raising a mixed number to a power finds applications in many real-world scenarios, including:

• Volume Calculations: As mentioned earlier, calculating the volume of a cube or other three-dimensional shapes with fractional side lengths.
• Compound Interest: Calculating compound interest over multiple periods, where the interest rate is expressed as a mixed number.
• Engineering and Physics: Many engineering and physics problems involve calculations with fractional quantities raised to powers.
• Data Analysis and Statistics: Calculations involving statistical measures sometimes require manipulating numbers in this way.

Understanding the principles behind such calculations is fundamental to success in these fields.

Troubleshooting Common Mistakes

A common mistake is incorrectly converting the mixed number to an improper fraction. Always ensure you correctly multiply the whole number by the denominator and then add the numerator before keeping the same denominator.

Another frequent error is forgetting the order of operations. Remember to perform the exponentiation before any other operations unless parentheses dictate otherwise. This is especially important when dealing with more complex equations that include addition, subtraction, multiplication, and division, along with exponents.

Conclusion: Mastering Exponents with Mixed Numbers

Raising a mixed number to a power, such as cubing 2 5/3, may seem challenging at first glance, but by breaking down the problem into manageable steps and understanding the underlying principles of mixed numbers and exponents, the process becomes significantly clearer. This article has provided two different methods to achieve this, along with practical applications and explanations of common errors. By mastering these techniques, you will develop a stronger foundation in mathematics and its applications in various fields. Remember to practice regularly to enhance your proficiency in handling such calculations. The ability to perform these calculations efficiently and accurately is a valuable skill in various mathematical and scientific contexts.