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Complex Exponents

Exponent of a variable is referred as the power which is raised to it. Exponentiation is defined as multiplication of a quantity (a number or a value or a variable) to itself. Exponents can be defined as:

Exponent of Variable
Here, n is exponent or power of the variable x.

Exponents can be written in form of complex quantities also. These quantities are known as complex exponents. Following types of exponential expressions are included among complex exponents:
  • Negative Exponents: As the name suggests, the exponents with negative sign are known as negative exponents. For example: $x^{-2}$.
  • Fractional Exponents or Rational Exponents: The exponents in the form of fractions or rational numbers, are known as fractional or rational exponents. For example: $x^{\frac{1}{4}}$,$8^{\frac{2}{3}}$ etc.
  • Radical Exponents: Radical exponents are the powers in the form of radicals. Radical is also referred as root. For example: $\sqrt[5]{x}, \sqrt[4]{16}$ etc. Second root of variable "x" is represented by $\sqrt{x}$ and is pronounced as square root of x. Similarly, its third root is represented by $\sqrt[3]{x}$ and is pronounced as cube root of x.
  • Recursive Exponents: It is the exponent with recursive powers, i.e., an exponent over an exponent. For example: $3^{2^{3}}$, $x^{x^{x^{x...}}}$ etc.
  • Complex (imaginary) Exponents: An exponent with imaginary number "i" or "iota", is called complex or imaginary exponent. For example: $e^{i\theta }$.
  • Mixed Exponents: Exponents which include more than one of above types of exponents. For example: $(256)^{-\frac{1}{4}}$, $\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}$ etc.
Example: Solve $\sqrt[5]{243}$.

Solution: $\sqrt[5]{243}$

= $(243)^{\frac{1}{5}}$

= $(3^{5})^{\frac{1}{5}}$

= $(3)^{5 \times \frac{1}{5}}$

= 3

Related Calculators
Calculating Exponents Adding Complex Fractions Calculator
Adding Complex Numbers Calculator Complex Fraction Calculator

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