Compute Displaystyle Left Frac 8 125 Right 1 3

Decoding the Calculation: A Deep Dive into (8/125)^(1/3)

This article provides a comprehensive explanation of how to solve the mathematical expression (8/125)^(1/3), delving into the underlying principles and demonstrating multiple approaches to arrive at the solution. We'll explore the concepts of exponents, fractions, and roots, ultimately guiding you to a clear understanding of this seemingly complex calculation. This detailed walkthrough is perfect for students tackling introductory algebra, as well as anyone looking to refresh their mathematical skills.

Understanding the Problem: Exponents and Fractional Exponents

The expression (8/125)^(1/3) involves two key mathematical concepts: exponents and fractional exponents. Let's break them down:

• Exponents: An exponent, also known as a power or index, indicates repeated multiplication. For example, 2³ means 2 × 2 × 2 = 8. The base (2) is multiplied by itself the number of times specified by the exponent (3).

• Fractional Exponents: A fractional exponent represents a combination of exponentiation and root extraction. The expression a^(m/n) is equivalent to the nth root of a raised to the power of m, or (ⁿ√a)ᵐ. In our case, (8/125)^(1/3) means the cube root (n=3) of (8/125) raised to the power of 1 (m=1). Since raising to the power of 1 doesn't change the value, we essentially need to find the cube root of 8/125.

Method 1: Simplifying the Fraction and then Finding the Cube Root

This method involves simplifying the fraction within the parentheses before tackling the cube root.

1. Simplify the Fraction: The fraction 8/125 can be simplified by finding the greatest common divisor (GCD) of 8 and 125. Since 8 = 2³ and 125 = 5³, there is no common divisor other than 1. Therefore, the fraction is already in its simplest form.

2. Find the Cube Root: Now, we need to find the cube root of 8/125. This means finding a number which, when multiplied by itself three times, equals 8/125. We can separately find the cube root of the numerator and the denominator:

   • Cube root of 8: 2 × 2 × 2 = 8, so the cube root of 8 is 2.
   • Cube root of 125: 5 × 5 × 5 = 125, so the cube root of 125 is 5.

3. Combine the Results: Therefore, the cube root of 8/125 is 2/5.

Method 2: Applying the Rules of Exponents

This approach leverages the properties of exponents to simplify the expression before calculating the numerical value.

1. Distribute the Exponent: We can rewrite the expression as (8^(1/3)) / (125^(1/3)). This is allowed because (a/b)^n = a^n / b^n.

2. Calculate the Cube Roots: Now, we can individually calculate the cube roots:

   • 8^(1/3) = 2
   • 125^(1/3) = 5

3. Combine the Results: The final answer is 2/5.

Method 3: Prime Factorization and Simplification

Prime factorization offers another effective way to solve this problem.

1. Prime Factorization: Express both 8 and 125 as their prime factors:

   • 8 = 2 × 2 × 2 = 2³
   • 125 = 5 × 5 × 5 = 5³

2. Substitute into the Expression: Replace 8 and 125 in the original expression with their prime factorizations: ((2³)/(5³))^(1/3)

3. Apply the Power of a Quotient Rule: (a/b)^n = a^n / b^n. This allows us to distribute the exponent (1/3) to both the numerator and the denominator: (2³)^(1/3) / (5³)^(1/3)

4. Simplify using Power of a Power Rule: (a^m)^n = a^(mn). This simplifies the expression to: 2^(3(1/3)) / 5^(3*(1/3)) = 2¹ / 5¹ = 2/5

Understanding the Solution: What Does 2/5 Represent?

The solution, 2/5, represents a fraction, a ratio of two integers. It also represents a decimal value of 0.4. This shows that the cube root of 8/125 is equal to 2/5 or 0.4.

Expanding the Concept: Other Fractional Exponents

The principles discussed above extend to other fractional exponents. For example, consider the expression (16)^(3/4). This can be solved as follows:

1. Rewrite as a Root and Power: (16)^(3/4) = (⁴√16)³

2. Calculate the Fourth Root: ⁴√16 = 2 (because 2 × 2 × 2 × 2 = 16)

3. Raise to the Power of 3: 2³ = 8

Therefore, (16)^(3/4) = 8.

Practical Applications and Real-World Examples

Fractional exponents find application in various fields, including:

• Physics: Calculating the volume of a sphere, the surface area of a cube or the velocity of falling objects.

• Engineering: Designing structures, analyzing stress and strain, and calculating power consumption.

• Finance: Compound interest calculations, determining the present value of future cash flows.

• Chemistry: Calculating reaction rates and concentrations.

Troubleshooting Common Mistakes

When working with fractional exponents, be mindful of the following potential pitfalls:

• Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).

• Negative Exponents: A negative exponent implies the reciprocal. For example, a^(-n) = 1/a^n.

• Zero Exponent: Any number (except 0) raised to the power of 0 equals 1.

Conclusion: Mastering Fractional Exponents

Understanding fractional exponents is crucial for mastering many mathematical and scientific concepts. By applying the principles discussed in this article, and practicing regularly, you can confidently tackle complex calculations and solve problems involving fractional exponents. Remember to break down complex problems into smaller, more manageable steps. Use the different methods demonstrated above to approach similar problems from various angles and reinforce your understanding. The more you practice, the more comfortable and proficient you'll become. This empowers you to confidently apply these skills in your studies and beyond.