# Arithmetic with Exponents

## Addition and Subtraction

Addition and subtraction with exponents work exactly the same way that they do with any other variable. If the variable parts are the same, you can add the expressions just by adding their numeric parts so

2*x*^{2}*y*^{3} - 5*x*^{2}*y*^{3} = (2 - 5)*x*^{2}*y*^{3} = -3*x*^{2}*y*^{3}

but

*z*^{3} + *x*^{3}

can't be simplified any further because the variable parts aren't *exactly* the same. The exponents are equal, three in both parts, but because the variables are different, *z* in one part and *x* in the other, there's nothing that can be done to combine the two.

## Multiplication

Two multiply two exponents *that have the same base* all you have to do is add their exponents. So

*x*^{2} · *x*^{4} = *x*^{2 + 4} = *x*^{6}

Remember that this rule only applies if the expressions are being multiplied. So *x*^{3} + *x*^{3} is equal to 2*x*^{3}, *not* *x*^{6}.

Remember that it isn't necessary for two terms in an expression to be next to each other for this to work. So, for example

*x*^{3}*y*^{3}*x*^{2} = *x*^{3 + 2}*y*^{3} = *x*^{5}*y*^{3}

Notice that I didn't try to combine the exponents any further. Because the bases are different, *y* in one part and *x* in the other, the expression is as simplified as it can be.

## Division

Two divide to exponents *that have the same base* you subtract their exponents. So, for example,

x^{5} |
= x^{5 - 2} = x^{3} |

x^{2} |

All the warning from the previous section apply here as well. First, the bases have to be the same. If the numerator in my previous example had a *y* instead of an *x* then there wouldn't have been anything we could have done to simplify it further. Second, there can't be any addition involved. For example,

x^{5} + y^{5} |
= x^{5 - 2} = x^{3} |

x^{2} |

can't be simplified any further. Specifically, you can't just subtract the exponents on the *x*'s and claim that the expression equals *x*^{3} + *y*^{5}. The only thing you could possibly do here would be to split the fraction in two

x^{5} + y^{5} |
= | x^{5} |
+ | y^{5} |

x^{2} |
x^{2} |
x^{2} |

Now you can reduce the part with *x*'s in both the numerator and denominator which would leave you with

x^{3}+ |
y^{5} |

x^{2} |

as your simplified version.