<< Chapter < Page | Chapter >> Page > |
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations , expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.
Simplify:
ⓐ | |
Notice the exponent applies to just the base $y$ . | $5{y}^{\mathrm{-1}}$ |
Take the reciprocal of $y$ and change the sign of the exponent. | $5\xb7\frac{1}{{y}^{1}}$ |
Simplify. | $\frac{5}{y}$ |
ⓑ | |
Here the parentheses make the exponent apply to the base $5y$ . | ${(5y)}^{\mathrm{-1}}$ |
Take the reciprocal of $5y$ and change the sign of the exponent. | $\frac{1}{{(5y)}^{1}}$ |
Simplify. | $\frac{1}{5y}$ |
ⓒ | |
${(\mathrm{-5}y)}^{\mathrm{-1}}$ | |
The base is $-5y$ . Take the reciprocal of $-5y$ and change the sign of the exponent. | $\frac{1}{{(\mathrm{-5}y)}^{1}}$ |
Simplify. | $\frac{1}{\mathrm{-5}y}$ |
Use $\frac{a}{-b}=-\frac{a}{b}.$ | $-\frac{1}{5y}$ |
Simplify:
Simplify:
Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},$ where $a\ne 0$ and m and n are integers.
When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, ${a}^{-n}=\frac{1}{{a}^{n}}.$
All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
If $a,b$ are real numbers and $m,n$ are integers, then
Simplify:
ⓐ | |
${x}^{\mathrm{-4}}\xb7{x}^{6}$ | |
Use the Product Property, ${a}^{m}\xb7{a}^{n}={a}^{m+n}.$ | ${x}^{\mathrm{-4}+6}$ |
Simplify. | ${x}^{2}$ |
ⓑ | |
${y}^{\mathrm{-6}}\xb7{y}^{4}$ | |
The bases are the same, so add the exponents. | ${y}^{\mathrm{-6}+4}$ |
Simplify. | ${y}^{\mathrm{-2}}$ |
Use the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}.$ | $\frac{1}{{y}^{2}}$ |
ⓒ | |
${z}^{\mathrm{-5}}\xb7{z}^{\mathrm{-3}}$ | |
The bases are the same, so add the exponents. | ${z}^{\mathrm{-5-3}}$ |
Simplify. | ${z}^{\mathrm{-8}}$ |
Use the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}.$ | $\frac{1}{{z}^{8}}$ |
Simplify:
Simplify:
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents .
Simplify: $({m}^{4}{n}^{\mathrm{-3}})({m}^{\mathrm{-5}}{n}^{\mathrm{-2}}).$
$({m}^{4}{n}^{\mathrm{-3}})({m}^{\mathrm{-5}}{n}^{\mathrm{-2}})$ | |
Use the Commutative Property to get like bases together. | ${m}^{4}{m}^{\mathrm{-5}}\xb7{n}^{\mathrm{-2}}{n}^{\mathrm{-3}}$ |
Add the exponents for each base. | ${m}^{\mathrm{-1}}\xb7{n}^{\mathrm{-5}}$ |
Take reciprocals and change the signs of the exponents. | $\frac{1}{{m}^{1}}\xb7\frac{1}{{n}^{5}}$ |
Simplify. | $\frac{1}{m{n}^{5}}$ |
Simplify: $({p}^{6}{q}^{\mathrm{-2}})({p}^{\mathrm{-9}}{q}^{\mathrm{-1}}).$
$\frac{1}{{p}^{3}{q}^{3}}$
Simplify: $({r}^{5}{s}^{\mathrm{-3}})({r}^{\mathrm{-7}}{s}^{\mathrm{-5}}).$
$\frac{1}{{r}^{2}{s}^{8}}$
If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation .
Simplify: $(2{x}^{\mathrm{-6}}{y}^{8})(\mathrm{-5}{x}^{5}{y}^{\mathrm{-3}}).$
$(2{x}^{\mathrm{-6}}{y}^{8})(\mathrm{-5}{x}^{5}{y}^{\mathrm{-3}})$ | |
Rewrite with the like bases together. | $2(\mathrm{-5})\xb7({x}^{\mathrm{-6}}{x}^{5})\xb7({y}^{8}{y}^{\mathrm{-3}})$ |
Simplify. | $\mathrm{-10}\xb7{x}^{\mathrm{-1}}\xb7{y}^{5}$ |
Use the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}.$ | $\mathrm{-10}\xb7\frac{1}{{x}^{1}}\xb7{y}^{5}$ |
Simplify. | $\frac{\mathrm{-10}{y}^{5}}{x}$ |
Simplify: $(3{u}^{\mathrm{-5}}{v}^{7})(\mathrm{-4}{u}^{4}{v}^{\mathrm{-2}}).$
$-\frac{12{v}^{5}}{u}$
Simplify: $(\mathrm{-6}{c}^{\mathrm{-6}}{d}^{4})(\mathrm{-5}{c}^{\mathrm{-2}}{d}^{\mathrm{-1}}).$
$-\frac{30{d}^{3}}{{c}^{8}}$
Notification Switch
Would you like to follow the 'Prealgebra' conversation and receive update notifications?