Exponent laws/rules are useful tools when evaluating expressions. They include the product rule, quotient rule, and power rules. Given integers $m$ and $n$, we have the following:
Product Rule
\[\begin{align*} x^m \cdot x^n = x^{m + n} \end{align*}\]
This works as if we expand the exponent out as multiplications of $x$, then $x^m = (x \cdot x \cdots x)$ $m$ times. Doing the same with $x^n$, if we multiply them, we get a string of $(x \cdot x \cdots x)$ $m + n$ times. This can be written as $x^{m + n}$.
Practice Problem – Evaluate $2^3 \cdot 2^7$.
Using our product rule, we can say $2^3 \cdot 2^7 = 2^{10} = 1024$.
Quotient Rule
\[\begin{align*} \dfrac{x^m}{x^n} = x^{m – n} \end{align*}\]
This works similarly to the product rule as expanding the $x^m = (x \cdot x \cdots x)$ $m$ times and $x^n$ the same way, we may cancel out the $x$’s on the numerator and denominator, leaving us with the difference of $m$ and $n$.
Practice Problem – Evaluate $\dfrac{2^7}{2^3}$.
Using the quotient rule, we get $\dfrac{2^7}{2^3} = 2^{7 – 3} = 2^4 = 16$.
Negative Exponents and Zero
Note, if $m > n$, then we are left with a positive integer as the exponent, but if $m < n$, we are left with a negative number in the exponent. A negative exponent is nothing to worry about though! Using the idea from above, if given integer $k$, then:
\[\begin{align*} x^{-k} = \dfrac{1}{x^k} \end{align*}\]
This comes from the fact that all the $x$’s in the numerator were canceled out, leaving us with just the ones left in the denominator. If $m – n = 0$, note that any non-zero integer to the power of $0$ is $1$.
Practice Problem – Evaluate $\dfrac{2^3}{2^7}$.
Using the quotient rule, we get $\dfrac{2^3}{2^7} = 2^{3 – 7} = 2^{-4} = \dfrac{1}{2^4} = \dfrac{1}{16}$.
Power Rule
\[\begin{align*} (x^m)^n = x^{m \cdot n} \end{align*}\]
Once again expanding this, we would have $n$ copies of $x^m$ expanded next to each other. This would be $(x \cdot x \cdots x)$ $m \cdot n$ times. Using this in conjunction with the product rule and quotient rule:
\[\begin{align*} (xy)^m &= x^my^m\\\\ \left(\dfrac{x}{y}\right)^m &= \dfrac{x^m}{y^m} \end{align*}\]
The exponent gets distributed to each component individually as these can always be written as $(x \cdot y)^m$ and $\left(x \cdot \dfrac1y\right)^m$ respectively. Expanding, we get $m$ copies of each, hence the rule holds.
Practice Problem – Simplify $(2^3)^7$.
Using the power rule, we get $(2^3)^7 = 2^{3 \cdot 7} = 2^{21}$.
Fractional Exponents
\[\begin{align*} x^{\frac{m}{n}} = \sqrt[n]{x^m} \end{align*}\]
Using the power rule, note that $(x^m)^\frac{1}{m} = x$. This means that $\dfrac{1}{m}$ in this case is the $m^{th}$ root of $x$. Applying this with integer $n$ instead of specifically $m$, we get that it would be the $n^{th}$ root of $x^m$.
Practice Problem – Simplify $2^{\frac{7}{3}}$.
Using the other rules and the information on fractional exponents, we get:
\[\begin{align*} 2^{\frac{7}{3}} = \sqrt[3]{x^7} \end{align*}\]
Summary
The product, quotient, negative, power, and fractional exponent rules are all listed below:
\[\begin{align*} x^m \cdot x^n &= x^{m + n}\\\\ \dfrac{x^m}{x^n} &= x^{m – n}\\\\ x^{-k} &= \dfrac{1}{x^k}\\\\ (xy)^m &= x^my^m\\\\ \left(\dfrac{x}{y}\right)^m &= \dfrac{x^m}{y^m}\\\\ x^{\frac{m}{n}} &= \sqrt[n]{x^m} \end{align*}\]
