To simplify the power of a product of two exponential expressions, we can use the
power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider
$\text{\hspace{0.17em}}{\left(pq\right)}^{3}.\text{\hspace{0.17em}}$ We begin by using the associative and commutative properties of multiplication to regroup the factors.
In other words,
$\text{\hspace{0.17em}}{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}.$
The power of a product rule of exponents
For any real numbers
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and any integer
$\text{\hspace{0.17em}}n,$ the power of a product rule of exponents states that
${\left(ab\right)}^{n}={a}^{n}{b}^{n}$
Using the power of a product rule
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
${\left(a{b}^{2}\right)}^{3}$
${\left(2t\right)}^{15}$
${\left(\mathrm{-2}{w}^{3}\right)}^{3}$
$\frac{1}{{\left(\mathrm{-7}z\right)}^{4}}$
${\left({e}^{\mathrm{-2}}{f}^{2}\right)}^{7}$
Use the product and quotient rules and the new definitions to simplify each expression.
To simplify the power of a quotient of two expressions, we can use the
power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
For any real numbers
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and any integer
$\text{\hspace{0.17em}}n,$ the power of a quotient rule of exponents states that
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.