Section 5.1: The Product and Power Rules for Exponents Worksheet 5.1
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 $x^{2}\cdot x^{4}$
Example: Simplify the expression $x^{21}\cdot x^{14}$
Example: Simplify the expression $-2x^{3}\cdot 3x^{9} \cdot (-3)x^{2}$
The Product Rule: $x^m \cdot x^n=x^{m+n}$
Example (a not-so-simple base): Simplify the expression
$$(-y+z)^{3}\cdot (-y+z)^{4}$$
The Power Rule: $(x^m)^n=x^{m \cdot n}$
Example: Simplify the expression $(x^{2})^{4}$
Example: Simplify the expression $(x^{21})^{14}$
Example: Simplify the expression $(-x^{2})^4$
Example: Simplify the expression $-(x^{2})^4$
Powers of Products and Quotients
Power of a Product: $(xy)^n=x^n y^n$
Example: Simplify $(xy)^3$
Power of a Quotient: $\left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}$
Example: Simplify $\left(\frac{x}{y}\right)^3$
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: $\left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}$
Simplify the following expressions.
$(r^4 \cdot r^6)^{2}$
$(-3t^4 \cdot \gamma^5)^{2}$
$-(\frac{4c^4}{x^5})^{2}$