Exponent
- Tags
- math
Refers to the multiplication of a number by itself a certain number of times. We say a number \( x \) raised to the power \( y \) is equivalent to \( x^y = x \times x \times \dots \times x = \prod^{y} x \).
The special case of
- \( y = 2 \) is known as squaring.
- \( y = 3 \) is known as cubing.
Rules of Powers
Raising to the power can have different affects depending on the value of \( x \) and \( y \).
Table 1:
Listing of possible \( x,y \) combinations and their equivalent forms.
| LHS | RHS | Example |
|---|---|---|
| \( x^0 \) | \( 1 \) | \( 5^0 = 1 \) |
| \( x^1 \) | \( x \) | \( 5^1 = 5 \) |
| \( (\frac{x}{z})^y \) | \( \frac{xy}{zy} \) | \( (\frac{1}{2})^2 = \frac{1}{4} \) |
| \( (xz)^y = \) | \( x^y z^y \) | \( (2 \times 5)^2 = (2^2 \times 5^2) = 100 \) |
| \( x^a x^b \) | \( x^{a+b} \) | \( x^2 x^4 = x^6 \) |
| \( x^a / x^b \) | \( x^{a-b} \) | \( x^6 / x^2 = x^4 \) |
| \( (xa)b \) | \( x^{ab} \) | \( (x3)4 = x^{12} \) |
| \( x^{-a} \) | \( \frac{1}{x^a} \) | \( x^{-2} = 1 / x^2 \) |
| \( x^{1 / a} \) | \( \sqrt{a}{x} \) | \( x^{1 / a} = \sqrt[2]{x} \) |
| \( x^{b / a} \) | \( \sqrt[a]{x^b} \) | \( x^{2 / 3} = \sqrt[3]{x^2} \) |