Section 5.3: Negative Exponents and Scientific Notation Worksheet 5.3
Negative Exponents
Negative exponents tell us that there are factors downstairs. That is,
$$x^{-n}=\frac{1}{x^n}$$
tells us that there are $n$ factors of $x$ in the denominator.
Example: $x^{-1}$
Example: $x^{-3}$
Example: $\left(\frac{1}{x}\right)^{-1}$
Example: $\left(\frac{1}{x}\right)^{-3}$
Example: $\frac{1}{x^{-1}}$
Example: $\frac{1}{x^{-3}}$
Example: $2^{-1}+3^{-1}$
Some General Rules for Negative Exponents
Upstairs/Downstairs 1: $\frac{1}{x^{-n}}=x^{n}$
Upstairs/Downstairs 2: $x^{-n}=\frac{1}{x^n}$ (This is our definition of negative exponent.)
Fraction Flip: $\left(\frac{x}{y}\right)^{-n}=\left(\frac{y}{x}\right)^{n}$
Example: Write the expression without negative exponents. $\frac{y^2}{zx^{-3}}$
Rules for Negative Exponents: They all hold. In fact, they're simpler!
The Product Rule: $x^m \cdot x^n=x^{m+n}$
The Quotient Rule: $\frac{x^m}{x^n}=x^{m-n}$
The Power Rule: $(x^m)^n=x^{m \cdot n}$
Power of a Product: $(xy)^n=x^n y^n$
Power of a Quotient: $\left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}$
Example:Simplify the expression by writing it without negative exponents.
$$\left(\frac{2y^2}{zx^{-3}}\right)^{-4}$$
More Examples: Simplify the following expressions by writing them without any fraction bars.
$\left(\frac{\theta^{7}}{c^{17}}\right)^{-11}$
$\frac{16p^{4}q^{3}}{4p^{5}q^{2}}$
$(y^{3} \cdot \theta^{-7})^{6}(y^{-4} \cdot \theta^{-2})^{5}$
$\left[\frac{(\phi^{-8} \cdot r^{3})^{2}(\phi^{-6} \cdot r^{-4})^{7}}{(\phi^{4} \cdot r^{5})^{-7}}\right]^{-3}$