# rules of exponents

## Quick guide

This is a summary of the rules of exponents material used in the xlf presentation series. It include numerical examples, and keystrokes for the Hewlett Packard 12C financial calculator.

HP 12C RPN keystrokes

### Exponents

A exponent is shown as a superscript to a base number – see figure 1.

Exponent rule Numerical example
$$x^0=1$$
See Note 1
$$5^0=1$$

$$\text{12c: }5\;[\text{Enter}]\;0\;[y^x]\;\text{returns} \;1$$
$$x^1=x$$ $$4^1=4$$

$$\text{12c: }4\;[\text{Enter}]\;1\;[y^x]\;\text{returns} \;4$$
$$x^{-1}=1/x$$ $$2^{-1}=1/2$$

$$\text{12c: }2\;[\text{Enter}]\;1\;[\text{CHS}]\;[y^x]\;\text{returns} \;0.5$$
Product rule
$$x^j x^k=x^{j+k}$$
$$x^2x^4=x^{2+4}=x^6$$
$$2^2\cdot2^4=4\cdot16=64; \quad 2^6=64$$

$$\text{12c: }2\;[\text{Enter}]\;2\;[y^x]\;2\;[\text{Enter}]\;4\;[y^x]\;[×]\;\text{returns}\;64$$
Quotient rule
$$x^j/x^k=x^{j-k}$$
$$x^7/x^3=x^{7-3}=x^4$$
$$5^7/5^3=5^{7-3}=5^4=625; \quad 78125/125=625$$

$$\text{12c: }5\;[\text{Enter}]\;7\;[y^x]\;5\;[\text{Enter}]\;3\;[y^x]\;[÷]\;\text{returns} \;625$$
Power rule
$$(x^j)^k=x^{jk}$$
$$(x^2)^4=x^{2\cdot4}=x^8$$
$$(4^2)^4=4^8=65536; \quad 16^4=65536$$

$$\text{12c: }4\;[\text{Enter}]\;2\;[y^x]\;\;4\;[y^x]\;\text{returns} \;65536$$
$$(xy)^j=x^jy^j$$ $$(xy)^3=x^3y^3$$
$$(2\cdot3)^3=2^3\cdot3^3=8\cdot27=216; \quad 6^3=216$$

$$\text{12c: }2\;[\text{Enter}]\;3\;[×]\;3\;[y^x]\;\text{returns}\;216$$
$$(x/y)^j=x^j/y^j$$ $$(x/y)^2=x^2/y^2$$
$$(4/5)^2=4^2/5^2=16/25=0.64; \quad(0.8)^2=0.64$$

$$\text{12c: }4\;[\text{Enter}]\;5\;[÷]\;2\;[y^x]\;\text{returns}\;0.64$$
$$x^{-j}=1/x^j$$ $$x^{-3}=1/x^3$$
$$2^{-3}=1/2^3=1/8; \quad2^{-3}=0.125$$

$$\text{12c: }2\;[\text{Enter}]\;3\;[\text{CHS}]\;[y^x]\;\text{returns}\;0.125$$

Note 1: $$\dfrac{x^j} {x^j}=1$$, thus by application of the quotient rule $$x^{j-j}=x^0=1$$

### The exponential function

The special case where the base $$x$$ is set to the mathematical constant $$e, 2.71828$$ is important in many areas of finance, such as interest rates in continuous time. The notation $$e^x$$ is called the exponential function, and is also written as $$exp(x)$$. From the exponent rules, $$exp(1)=e=2.71828$$, and $$exp(0)=1$$.