Simplify The Exponential Expression 48 Superscript One-half

Simplifying the Exponential Expression 48^1/2: A Comprehensive Guide

Understanding and simplifying exponential expressions is a fundamental skill in mathematics, crucial for various applications in science, engineering, and finance. This article provides a detailed exploration of simplifying the expression 48^1/2, covering the underlying concepts, step-by-step solutions, and practical applications. We'll delve into the meaning of fractional exponents, prime factorization, and the simplification process, aiming for a thorough understanding suitable for students and anyone looking to refresh their mathematical skills.

Understanding Fractional Exponents

Before tackling 48^1/2, let's establish a firm grasp of fractional exponents. A fractional exponent, such as the "^1/2" in our expression, represents a combination of two operations: exponentiation and root extraction. The numerator of the fraction indicates the power to which the base is raised, and the denominator indicates the root to be taken.

Specifically, a^m/n = (ⁿ√a)^m. This means that "a raised to the power of m/n" is equivalent to taking the nth root of 'a' and then raising the result to the power of 'm'.

In the case of 48^1/2, 'a' is 48, 'm' is 1, and 'n' is 2. Therefore, 48^1/2 is equivalent to the square root of 48: √48.

Prime Factorization: The Key to Simplification

The next crucial step in simplifying √48 is to perform prime factorization. Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are numbers greater than 1 that have only two divisors: 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).

Let's factorize 48:

48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3

Therefore, 48 can be expressed as 2⁴ x 3.

Simplifying the Square Root

Now, we can substitute the prime factorization back into our original expression:

√48 = √(2⁴ x 3)

Using the property of square roots that √(a x b) = √a x √b, we can separate the square root:

√(2⁴ x 3) = √2⁴ x √3

Since √2⁴ is equivalent to (2⁴)^1/2, we can apply the power rule of exponents: (a^m)ⁿ = a^mn. This simplifies to:

(2⁴)^1/2 = 2^{4 x (1/2)} = 2² = 4

Therefore:

√48 = 4√3

The Final Simplified Expression

By combining the steps above, we've successfully simplified 48^1/2:

48^1/2 = √48 = 4√3

This is the simplest form of the expression, as 3 is a prime number and cannot be further simplified within the square root.

Practical Applications and Further Exploration

Understanding the simplification of exponential expressions like 48^1/2 has numerous practical applications across various fields:

• Geometry: Calculating the length of a diagonal in a rectangle or the hypotenuse of a right-angled triangle often involves simplifying square roots. For instance, finding the length of the diagonal of a square with sides of length 4√3 units would directly utilize this knowledge.

• Physics: Many physical formulas incorporate square roots, particularly in areas such as kinematics (dealing with motion), mechanics (dealing with forces and motion), and electromagnetism.

• Engineering: Similar to physics, engineering calculations frequently involve simplifying square roots and other radical expressions to obtain precise measurements and calculations.

• Finance: Compound interest calculations often involve exponential expressions, and simplifying these expressions allows for easier understanding and interpretation of financial data.

Exploring Higher-Order Roots

The principles discussed here extend beyond square roots (^1/2). We can apply the same techniques to higher-order roots, such as cube roots (^1/3), fourth roots (^1/4), and so on. The core concept remains the same: prime factorization is key to simplifying the expression. For instance, to simplify 125^1/3 (the cube root of 125), we would first find the prime factorization of 125 (5³) and then apply the rule for fractional exponents.

Working with Negative Exponents

Fractional exponents can also be negative. A negative fractional exponent indicates both a root and a reciprocal. For example, 48^-1/2 is equivalent to 1/√48, which we can further simplify using the methods described above.

Conclusion: Mastering Exponential Simplification

Simplifying the exponential expression 48^1/2 involves a methodical approach that combines understanding fractional exponents, prime factorization, and the properties of square roots. Mastering this process strengthens your mathematical foundation and provides essential tools for tackling more complex problems across various disciplines. By consistently practicing prime factorization and applying the rules of exponents, you can build confidence and proficiency in simplifying similar expressions, paving the way for more advanced mathematical endeavors. Remember, the key lies in breaking down complex expressions into their simplest forms through a systematic approach.