64 To The Power Of 2/3

64 to the Power of 2/3: A Deep Dive into Exponents and Roots

Understanding exponents and roots is fundamental to mathematics and numerous applications across various fields. This article delves into the calculation of 64 to the power of 2/3, exploring the underlying concepts, different approaches to solving it, and practical examples of its use. We'll break down the problem step-by-step, explaining the logic and techniques involved. Our goal is to provide a comprehensive understanding not just of this specific calculation, but also the broader mathematical principles it embodies.

What Does 64^2/3 Mean?

The expression 64^2/3 represents 64 raised to the power of 2/3. This fractional exponent combines two fundamental mathematical operations: exponentiation and root extraction. Let's dissect the meaning:

• The Numerator (2): This indicates that we need to square the base (64). In simpler terms, we'll multiply 64 by itself.

• The Denominator (3): This represents the cube root. We need to find the number which, when multiplied by itself three times, equals 64.

Therefore, 64^2/3 can be interpreted in two equivalent ways:

1. (64^1/3)²: First, find the cube root of 64, and then square the result.

2. (64²)^1/3: First, square 64, and then find the cube root of the result.

Both methods yield the same answer, showcasing the flexibility of working with fractional exponents.

Method 1: Cube Root First, Then Square

This method prioritizes calculating the root before the exponent.

Step 1: Find the Cube Root of 64

The cube root of 64 (represented as 64^1/3) is the number that, when multiplied by itself three times, equals 64. That number is 4, because 4 x 4 x 4 = 64.

Step 2: Square the Result

Now, we square the result from Step 1: 4² = 4 x 4 = 16

Therefore, 64^2/3 = 16

Method 2: Square First, Then Cube Root

This alternative method prioritizes squaring before finding the root.

Step 1: Square 64

First, we square 64: 64² = 64 x 64 = 4096

Step 2: Find the Cube Root of the Result

Next, we find the cube root of 4096 (represented as 4096^1/3). This means finding the number which, when multiplied by itself three times, equals 4096. That number is 16, because 16 x 16 x 16 = 4096.

Therefore, 64^2/3 = 16

Why Both Methods Give the Same Result?

The equivalence of these two methods stems from the properties of exponents. Specifically, the power of a power rule: (a^m)ⁿ = a^mn. This rule demonstrates that multiplying exponents is equivalent to raising a power to another power. In our case:

(64^1/3)² = 64^{(1/3) * 2} = 64^2/3

(64²)^1/3 = 64^{2 * (1/3)} = 64^2/3

Both expressions simplify to 64^2/3, proving the equivalence of the two methods.

Understanding Fractional Exponents

Fractional exponents provide a concise way to represent both exponentiation and root extraction in a single operation. The general form is a^m/n, where:

• 'a' is the base.
• 'm' is the exponent (power).
• 'n' is the root (index of the radical).

This can be interpreted as: (a^1/n)^m or (a^m)^1/n

This means taking the nth root of 'a' and then raising it to the power of 'm', or raising 'a' to the power of 'm' and then taking the nth root of the result.

Applications of Fractional Exponents

Fractional exponents appear frequently in various scientific and mathematical contexts, including:

• Physics: Calculating the volume or surface area of objects with varying dimensions.

• Chemistry: Determining reaction rates and concentrations.

• Engineering: Analyzing stress, strain, and other material properties.

• Finance: Compound interest calculations.

• Calculus: Derivative and integral calculations often involve fractional exponents.

Solving Similar Problems

Let's apply the knowledge gained to solve similar problems:

Example 1: Calculate 8^2/3

1. Cube root of 8: 8^1/3 = 2 (since 2 x 2 x 2 = 8)
2. Square the result: 2² = 4

Therefore, 8^2/3 = 4

Example 2: Calculate 27^2/3

1. Cube root of 27: 27^1/3 = 3 (since 3 x 3 x 3 = 27)
2. Square the result: 3² = 9

Therefore, 27^2/3 = 9

Example 3: Calculate 125^2/3

1. Cube root of 125: 125^1/3 = 5 (since 5 x 5 x 5 = 125)
2. Square the result: 5² = 25

Therefore, 125^2/3 = 25

Negative Fractional Exponents

It's important to note that fractional exponents can also be negative. A negative fractional exponent, such as a^-m/n, implies both a reciprocal and a fractional exponent. It can be expressed as:

1/a^m/n

This means taking the reciprocal of a^m/n.

Example: Calculate 8^-2/3

1. Calculate 8^2/3: As we've seen, 8^2/3 = 4
2. Take the reciprocal: 1/4

Therefore, 8^-2/3 = 1/4

Conclusion

Calculating 64 to the power of 2/3, or 64^2/3, provides a practical illustration of how fractional exponents combine exponentiation and root extraction. Understanding this concept is crucial for navigating a wide range of mathematical and scientific problems. By mastering the techniques described, you'll be better equipped to tackle more complex calculations and deepen your comprehension of exponential and root operations. Remember the key steps: identify the base, the numerator (power), and the denominator (root), then apply the calculations sequentially (either root first, then power, or vice versa). The outcome will always be the same due to the fundamental laws of exponents. This understanding extends to negative fractional exponents, which simply add a reciprocal component to the calculation. The applications are vast, spanning various fields, highlighting the significance of this seemingly simple mathematical concept.