Exponentiation and its Calculation
Exponentiation, the mathematical operation of raising a number (the base) to a power (the exponent), extends beyond integer exponents to include fractional exponents. A fractional exponent represents a combination of exponentiation and root extraction.
Understanding Fractional Exponents
A fractional exponent of the form am/n is equivalent to taking the nth root of a raised to the power of m. This can be expressed as: (am)(1/n) or equivalently as: n√(am).
Calculator Methods
Using the Exponent Function (xy or ^)
Most calculators have an exponent function, often denoted as xy or ^. To calculate am/n, enter the base (a), then use the exponent function to raise it to the power of the fraction (m/n). Ensure the fraction is entered correctly, using parentheses if necessary to group the numerator and denominator.
Using Root Functions
Some calculators offer specific root functions (√, ³√, etc.). For am/n, one could calculate am first, then apply the appropriate nth root function. This approach is especially useful for simpler fractional exponents like 1/2 (square root) or 1/3 (cube root).
Using Logarithms (Advanced)
Logarithms provide an alternative method. The equation am/n = x can be solved using logarithms: log(x) = (m/n) log(a). This method requires solving for x, often involving antilogarithms (raising 10 or e to the power of the result). This method is less practical for direct calculation on most standard calculators.
Order of Operations
Following standard order of operations (PEMDAS/BODMAS) is crucial. Parentheses should be used to ensure correct calculation of the fractional exponent, especially when dealing with complex expressions involving other operations.
Example
To calculate 82/3:
- Using the exponent function: Enter 8, then press the xy (or ^) button, then enter (2/3), and press equals.
- Using root function: Calculate 82 (64), then take the cube root (³√64) to obtain the result.
The result in both cases should be 4.