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Exponentiation is a mathematical operation involving raising a base to an exponent. The operation is defined for positive integer exponents as repeated multiplication of the base by itself. In particular:

5^3=\underbrace{5\times5\times5}_{3\text{ times}}=125
a^5=\underbrace{a\times a\times a\times a\times a}_{5\text{ times}}
3^n=\underbrace{3\times3\times3\cdots\times3\times3}_{n\text{ times}}
a^b=\underbrace{a\times a\times a\cdots\times a\times a}_{b\text{ times}}

In the last example above, a is the base and b the exponent (or, less formally, power).

Other exponents

Negative exponents

A negative exponent is defined as one over the base to the power of the exponent. For instance,

x^{-2}=\frac1{x^2}

Fractional exponents

A fractional exponent is the value of the denominator root of the base to the power of the numerator. For example:

\begin{align}x^\frac12&=\sqrt{x}\\x^\frac23&=(\sqrt[3]{x})^2\end{align}

Common exponents

Square

"Squaring" a value means raising it to an exponent of 2, which is merely multiplying the value by itself.

a^2=a\times a

This term, and this simple case of exponentiation, holds a great deal of practicality and implications throughout basic mathematical concepts. Its commonality is emphasized by the fact that many calculators have a "square" button, but not necessarily the capacity for generalized exponentiation.

The term comes from the geometric figure it is named after. The area of any square is the length of its side squared.

Cube

"Cubing" a value means raising it to an exponent of 3. Again, the term comes originates from the geometric figure. The volume of a cube is equal to the length of a side cubed.

a^3=a\times a\times a

Evaluating exponents

Manual - Square roots

Modern calculators and computers allow determining square roots without difficulty. Before their invention, they needed to be done by hand, sometimes to a few decimal points.

For a square root, the number is divided into groups of two digits. In the first group, find the highest number who's square does not exceed the group, then subtract (similar to long division)

    7
 \/56 67
   49
    7 67

The next digit is more complicated. In general, you need to concatinate a test digit with twice of the quotient, then multipliy by the test digit. Again, you need the result to be less than the total.

    7  5
 \/56 67
   49
   --
    7 67
    7 25  (7*2 = 14, 145*5 = 725)
    ----
      42

This may be repeated past the decimal point, for additional digits (if desired).

    7  5. 2
 \/56 67 00
    -------
      42 00
      30 04
      -----
      11 96

Computer - squaring

The squaring method works for integer exponents.

 x^n=     
    \begin{cases}
                x \, ( x^{2})^{\frac{n - 1}{2}}, & \mbox{if } n \mbox{ is odd} \\
                (x^{2})^{\frac{n}{2}} , & \mbox{if } n \mbox{ is even}.
     \end{cases}



External links

  • Cube root - explains process for evaluating a cube root without a calculator.

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