Fractional Exponent Calculator

Unlock the mystery of fractional exponents! Input the base, numerator, and denominator to calculate the result and see the steps.

Result:

Calculation Steps:

Understanding Fractional Exponents

Fractional exponents are a way to express both powers and roots in a single notation. A fractional exponent is of the form m⁄n, where 'm' is the numerator and 'n' is the denominator.

Formula:

The general form of a fractional exponent is represented mathematically as:

$$ a^{{\frac{m}{n}}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m $$
  • \( a \) is the base.
  • \( m \) is the numerator, representing the power to which the base is raised.
  • \( n \) is the denominator, indicating the root to be taken.

Example:

Let's take an example to understand it better: Calculate \( 8^{\frac{2}{3}} \).

  • First, identify the base, numerator, and denominator:
    • Base (a) = 8
    • Numerator (m) = 2
    • Denominator (n) = 3
  • Apply the formula: \( 8^{\frac{2}{3}} = \sqrt[3]10 \) or \( (\sqrt[3]8)^2 \)
  • Calculate \( 8^2 = 64 \). Then find the cube root of 64: \( \sqrt[3]64 = 4 \)
  • Alternatively, find the cube root of 8 first: \( \sqrt[3]8 = 2 \). Then square the result: \( 2^2 = 4 \)
  • Both methods give the same result: \( 8^{\frac{2}{3}} = 4 \)

Use Cases:

Fractional exponents are used in various fields including:

  • Algebra and Calculus
  • Physics and Engineering
  • Computer Graphics and more