Recall: An exponent counts how many times a number is multiplied by itself.
That is, EXPONENTS COUNT REPEATED FACTORS.
Example: $2^3=2 \cdot 2 \cdot 2 =8$
Example: $x^3=x \cdot x \cdot x$
The Product Rule: $x^m \cdot x^n=x^{m+n}$
If we have $m$ factors of $x$ and $n$ factors of $x$, how many total factors of $x$ do we have?
Example: Simplify the expression $x^{2}\cdot x^{4}$
The long way: $x^{2}\cdot x^{4}=\underbrace{x \cdot x}_{2 \mbox{ copies}}\cdot\underbrace{x \cdot x \cdot x \cdot x}_{4 \mbox{ copies}}=\underbrace{x \cdot x \cdot x \cdot x \cdot x \cdot x}_{6 \mbox{ copies}}=x^6$
The shortcut way: $x^{2}\cdot x^{4}=x^{2+4}=x^6$
The shortcut way: $x^{2}\cdot x^{4}=x^{2+4}=x^6$
Example: Simplify the expression $x^{21}\cdot x^{14}$
$x^{21}\cdot x^{14}=x^{21+14}=x^{35}$
Example: Simplify the expression $-2x^{3}\cdot 3x^{9} \cdot (-3)x^{2}$
$$
\begin{array}{lll}
\displaystyle -2x^{3}\cdot 3x^{9} \cdot (-3)x^{2}&\displaystyle=-2\cdot(-3)\cdot 3\cdot x^{3}\cdot x^{9} \cdot x^{2} &\mbox{using commutative property}\\
\displaystyle &\displaystyle=6\cdot 3\cdot x^{3+9+2}&\mbox{using product rule}\\
\displaystyle &\displaystyle=18 x^{14}&\mbox{}\\
\end{array}
$$
The Product Rule: $x^m \cdot x^n=x^{m+n}$
Example (a not-so-simple base): Simplify the expression $$(-y+z)^{3}\cdot (-y+z)^{4}$$
$$
\begin{array}{lll}
\displaystyle (-y+z)^{3}\cdot (-y+z)^{4}&\displaystyle= (-y+z)^{3+4}&\mbox{}\\
\displaystyle &\displaystyle= (-y+z)^7&\mbox{}\\
\end{array}
$$
The Power Rule: $(x^m)^n=x^{m \cdot n}$
Example: Simplify the expression $(x^{2})^{4}$
The long way: $(x^{2})^{4}=\underbrace{x^2\cdot x^2 \cdot x^2 \cdot x^2}_{4 \mbox{ copies of } x^2}=\underbrace{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x }_{8 \mbox{ copies of } x}=x^8$
The shortcut way: $(x^{2})^{4}=x^{2\cdot 4}=x^8$
The shortcut way: $(x^{2})^{4}=x^{2\cdot 4}=x^8$
Example: Simplify the expression $(x^{21})^{14}$
$(x^{21})^{14}=x^{21\cdot 14}=x^{294}$
Example: Simplify the expression $(-x^{2})^4$
$$
\begin{array}{lll}
\displaystyle (-x^{2})^4&\displaystyle=\underbrace{(-x^{2})\cdot(-x^{2})\cdot(-x^{2})\cdot(-x^{2})}_{4 \mbox{ copies of } -x^{2}} &\mbox{}\\
\displaystyle &\displaystyle=\underbrace{(-1)\cdot x^{2}\cdot(-1)\cdot x^{2}\cdot(-1)\cdot x^{2}\cdot(-1)\cdot x^{2}}_{4 \mbox{ copies of } (-1)\cdot x^{2}} &\mbox{}\\
\displaystyle &\displaystyle=(-1)\cdot(-1)\cdot(-1)\cdot(-1)\cdot x^{2}\cdot x^{2}\cdot x^{2}\cdot x^{2}&\mbox{by commutative property}\\
\displaystyle &\displaystyle=1\cdot1\cdot x^{2+2+2+2}&\mbox{by product rule}\\
\displaystyle &\displaystyle= x^{8}&\mbox{}\\
\end{array}
$$
Okay. This one was a bit rough. But another shortcut will make things easier.
Example: Simplify the expression $-(x^{2})^4$
$$
\begin{array}{lll}
\displaystyle -(x^{2})^4&\displaystyle= -x^{2\cdot 4}&\mbox{by the power rule}\\
\displaystyle &\displaystyle= -x^8&\mbox{}\\
\end{array}
$$
Power of a Product: $(xy)^n=x^n y^n$
A power of a product is the product of the powers.
Example: Simplify $(xy)^3$
The long way: $(xy)^3=\underbrace{(xy)\cdot(xy)\cdot(xy)}_{3 \mbox{ copies of } xy}=x\cdot y\cdot x\cdot y \cdot x \cdot y=x\cdot x\cdot x\cdot y \cdot y \cdot y=x^3 y^3$
The shortcut way: $(xy)^3=x^3y^3$
The shortcut way: $(xy)^3=x^3y^3$
Example: Simplify $(2a)^4$
$(2a)^4=2^4 \cdot a^4=16a^4$
Example: Simplify $(-x^2)^4.$ (Remember this one?)
$$
\begin{array}{lll}
\displaystyle (-x^2)^4&\displaystyle=((-1)\cdot x^2)^4 &\mbox{rewrite as product}\\
\displaystyle &\displaystyle=(-1)^4 (x^2)^{4} &\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=1\cdot x^{2\cdot 4} &\mbox{by the power rule}\\
\displaystyle &\displaystyle=x^{8} &\mbox{}\\
\end{array}
$$
Power of a Quotient: $\displaystyle \left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}$
A power of a quotient is the quotient of the powers.
Example: Simplify $\displaystyle \left(\frac{x}{y}\right)^3$
The long way: $\displaystyle \left(\frac{x}{y}\right)^3=\underbrace{\frac{x}{y}\cdot \frac{x}{y}\cdot \frac{x}{y}}_{3 \mbox{ copies of } \frac{x}{y}}=\frac{x\cdot x \cdot x}{y\cdot y \cdot y}=\frac{x^3}{y^3}$
The shortcut way: $\displaystyle \left(\frac{x}{3}\right)^2=\frac{x^3}{y^3}$
The shortcut way: $\displaystyle \left(\frac{x}{3}\right)^2=\frac{x^3}{y^3}$
Example: Simplify $\displaystyle \left(\frac{3}{5}\right)^2$
The long way: $\displaystyle \left(\frac{3}{5}\right)^2=\frac{3}{5}\cdot \frac{3}{5}=\frac{3\cdot 3}{5\cdot 5}=\frac{9}{25}$
The shortcut way: $\displaystyle \left(\frac{3}{5}\right)^2=\frac{3^2}{5^2}=\frac{9}{25}$
The shortcut way: $\displaystyle \left(\frac{3}{5}\right)^2=\frac{3^2}{5^2}=\frac{9}{25}$
Example: Simplify $\displaystyle \left(\frac{2}{a}\right)^4$
$\displaystyle \left(\frac{2}{a}\right)^4=\frac{2^4}{a^4}=\frac{16}{a^4}$
Mixing it Up: Using Multiple Rules Simultaneously
The Product Rule: $x^m \cdot x^n=x^{m+n}$
The Power Rule: $(x^m)^n=x^{m \cdot n}$
Power of a Product: $(xy)^n=x^n y^n$
Power of a Quotient: $\displaystyle \left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}$
Simplify the following expressions.
$(r^4 \cdot r^6)^{2}$
$$
\begin{array}{lll}
\displaystyle (r^4 \cdot r^6)^{2}&\displaystyle= (r^4)^{2}\cdot(r^6)^{2}&\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle= r^{4\cdot 2}\cdot r^{6\cdot 2}&\mbox{Power Rule}\\
\displaystyle &\displaystyle= r^{8}\cdot r^{12}&\mbox{}\\
\displaystyle &\displaystyle= r^{8+12}&\mbox{Product Rule}\\
\displaystyle &\displaystyle= r^{20}&\mbox{}\\
\end{array}
$$
$(-3t^4 \cdot s^5)^{2}$
$$
\begin{array}{lll}
\displaystyle (-3t^4 \cdot s^5)^{2}&\displaystyle=(-3)^2 (t^4)^2 (s^5)^{2} &\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=9 t^{4\cdot 2} s^{5\cdot 2} &\mbox{Power Rule}\\
\displaystyle &\displaystyle=9 t^{8} s^{10} &\mbox{}\\
\end{array}
$$
$\displaystyle -\left(\frac{4c^4}{x^5}\right)^{2}$
$$
\begin{array}{lll}
\displaystyle -\left(\frac{4c^4}{x^5}\right)^{2}&\displaystyle=-\frac{(4c^4)^{2}}{(x^5)^{2}} &\mbox{A power of a quotient is the quotient of the powers!}\\
\displaystyle &\displaystyle=-\frac{4^{2}(c^4)^{2}}{(x^5)^{2}} &\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=-\frac{16c^{4\cdot 2}}{x^{5\cdot 2}} &\mbox{Power Rule}\\
\displaystyle &\displaystyle=-\frac{16c^{8}}{x^{10}} &\mbox{}\\
\end{array}
$$