Algebra: Arithmetic with Variables
Calculating with powers
The calculation rules for powers of numbers also apply to powers of variables:
Rules of calculation for powers Let \(a\) and \(b\) variables and \(r\) and \(s\) rational numbers. Then the following rules apply.
\[ \begin{array}{rcl}a^r\cdot a^s &=& a^{r+s}\\ \frac{a^r}{a^s}&=& a^{r- s}\\ (a^r)^s&=&a^{r\cdot s}\\ (a\cdot b)^r &=& a^r\cdot b^r\\ \frac{a^r}{b^r}&=& \left(\frac{a}{b}\right)^r\end {array}\]
The above calculation rules are formulated in 'elementary' form: the variables \(a\) and \(b\) can be representatives for all kinds of algebraic expressions.
If, for instance, #a# is replaced by #1+a^2# in the second equality, we find \[\frac{(1+a^2)^r}{(1+a^2)^s}= (1+a^2)^{r-s}\tiny.\]
Undefined expressions
Each variable can be replaced by any real number. Here are some exceptions:
- We cannot divide by #0#, for example, if #a=0# in #\frac{1}{a}# , then we have a problem.
- We cannot take the root of a negative number (see the discussion in Roots of integers), so if we take #a=-1# and #r=\frac{1}{2}# in #a^r#, then we have a problem.
We say that these terms are not defined or undefined for the chosen values of the variables.
Remember that \(y=y^1\) and apply the rule \(y^r\cdot y^s=y^{r+s}\), for rational numbers \(r\) and \(s\). This results in \[y^4\cdot y^{5}= y^{4+5}= y^{9}\tiny.\]
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