Section 1
Simplifying Expressions with Integral Exponents
Solve the resulting problems if the given changes are made in the indicated examples of this section.In Example $3,$ change the factor $x^{2}$ to $x^{-2}$ and then find the result.
Solve the resulting problems if the given changes are made in the indicated examples of this section.In Example $6,$ change the term $2 a$ to $2 a^{-1}$ and then find the result.
Solve the resulting problems if the given changes are made in the indicated examples of this section.In Example $8(b),$ change the $3^{-1}$ in the denominator to $3^{-2}$ and then find the result.
Solve the resulting problems if the given changes are made in the indicated examples of this section.In Example $9,$ change the sign in the numerator from $-$ to $+$ and then find the result.
Express each of the given expressions in simplest form with only positive exponents.$$x^{8} x^{-3}$$
Express each of the given expressions in simplest form with only positive exponents.$$y^{9} y^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$2 a^{2} a^{-6}$$
Express each of the given expressions in simplest form with only positive exponents.$$5 s s^{-5}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{c^{7}}{c^{-2}}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{t^{-8}}{t^{-3}}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{x^{5}}{y^{-2}}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{n^{-6}}{n n^{-4}}$$
Express each of the given expressions in simplest form with only positive exponents.$$5^{0} \times 5^{-3}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(3^{2} \times 4^{-3}\right)^{3}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(2 \pi x^{-1}\right)^{2}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(3 x y^{-2}\right)^{3}$$
Express each of the given expressions in simplest form with only positive exponents.$$2\left(5 a n^{-2}\right)^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$4\left(6 s^{2} t^{-1}\right)^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$(-4)^{0}$$
Express each of the given expressions in simplest form with only positive exponents.$$-4^{0}$$
Express each of the given expressions in simplest form with only positive exponents.$$-9 x^{0}$$
Express each of the given expressions in simplest form with only positive exponents.$$(-7 x)^{0}$$
Express each of the given expressions in simplest form with only positive exponents.$$3 x^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$(3 x)^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(7 a^{-1} x\right)^{-3}$$
Express each of the given expressions in simplest form with only positive exponents.$$7 a^{-1} x^{-3}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(\frac{2}{n^{3}}\right)^{-4}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(\frac{x^{3}}{-3}\right)^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$3\left(\frac{a}{b^{-2}}\right)^{-3}$$
Express each of the given expressions in simplest form with only positive exponents.$$5\left(\frac{2 n^{-2}}{D^{-1}}\right)^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$(a+b)^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$a^{-1}+b^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$2 x^{-3}+4 y^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$(3 x-2 y)^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(2 a^{-n}\right)^{2}\left(\frac{3}{2 a^{n}}\right)^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(7 \times 3^{-a}\right)\left(\frac{3^{a}}{7}\right)^{2}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(\frac{3 a^{2}}{4 b}\right)^{-3}\left(\frac{4}{a}\right)^{-5}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(2 n p^{-2}\right)^{-2}\left(4^{-1} p^{2}\right)^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(\frac{V^{-1}}{2 t}\right)^{-2}\left(\frac{t^{2}}{V^{-2}}\right)^{-3}$$
Express each of the given expressions in simplest form with only positive exponents.$$a b\left(\frac{a^{-2}}{b^{2}}\right)^{-3}\left(\frac{a^{-3}}{b^{5}}\right)^{2}$$
Express each of the given expressions in simplest form with only positive exponents.$$3 a^{-2}+\left(3 a^{-2}\right)^{4}$$
Express each of the given expressions in simplest form with only positive exponents.$$3\left(a^{-1} z^{2}\right)^{-3}+c^{-2} z^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$2 \times 3^{-1}+4 \times 3^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$5 \times 8^{-2}-3^{-1} \times 2^{3}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(R_{1}^{-1}+R_{2}^{-1}\right)^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$6^{-2}\left(2 a-b^{-2}\right)^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$\left(n^{-2}-2 n^{-1}\right)^{2}$$
Express each of the given expressions in simplest form with only positive exponents.$$2\left(3^{-3}-9^{-1}\right)^{-2}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{6^{-1}}{4^{-2}+2}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{x-y^{-1}}{x^{-1}-y}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{x^{-2}-y^{-2}}{x^{-1}-y^{-1}}$$
Express each of the given expressions in simplest form with only positive exponents.$$\frac{a x^{-2}+a^{-2} x}{a^{-1}+x^{-1}}$$
Express each of the given expressions in simplest form with only positive exponents.$$2 t^{-2}+t^{-1}(t+1)$$
Express each of the given expressions in simplest form with only positive exponents.$$(D-1)^{-1}+(D+1)^{-1}$$
Express each of the given expressions in simplest form with only positive exponents.$$4(2 x-1)(x+2)^{-1}-(2 x-1)^{2}(x+2)^{-2}$$
Solve the given problems.If $x<0,$ is it ever true that $x^{-2}<x^{-1}$ ?
Solve the given problems.Is it true that $(a+b)^{0}=1$ for all values of $a$ and $b ?$
Solve the given problems.Express $4^{2} \times 64$ (a) as a power of 4 and (b) as a power of 2
Solve the given problems.Express $1 / 81(\mathrm{a})$ as a power of 9 and $(\mathrm{b})$ as a power of 3.
Solve the given problems.(a) By use of Eqs. (11.4) and $(11.6),$ show that $\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^{n}$(b) Verify the equation in part (a) by evaluating each side with $a=3.576, b=8.091,$ and $n=7$
Solve the given problems.Evaluate $\left(8^{19}\right)^{12} /\left(8^{16}\right)^{14} .$ What happens when you try to evaluate this on a calculator?
Solve the given problems.For what integer value(s) of $n$ is $n^{\pi}>\pi^{n} ?$
Solve the given problems.Is it true that $\left[-2^{0}-(-1)^{0}\right]^{0}=1 ?$
Solve the given problems.Is it true that, if $x \neq 0,\left[\left(-x^{-2}\right)^{-2}\right]^{-2}=1 / x^{2} ?$
Solve the given problems.If $f(x)=4^{x},$ find $f(a+2).$
Solve the given problems.If $f(x)=2\left(9^{x}\right),$ find $f(2-a).$
Solve the given problems.Solve for $x: 2^{5 x}=2^{7}\left(2^{2 x}\right)^{2}.$
Solve the given problems.In analyzing the tuning of an electronic circuit, the expression $\left[\omega \omega_{0}^{-1}-\omega_{0} \omega^{-1}\right]^{2}$ is used. Expand and simplify this expression.
Solve the given problems.The metric unit of energy, the joule ( $\mathrm{J}$ ), can be expressed as $\mathrm{kg} \cdot \mathrm{s}^{-2} \cdot \mathrm{m}^{2} .$ Simplify these units and include newtons (see Appendix B) and only positive exponents in the final result.
Solve the given problems.The units for the electric quantity called permittivity are $\mathrm{C}^{2} \cdot \mathrm{N}^{-1} \cdot \mathrm{m}^{-2} .$ Given that $1 \mathrm{F}=1 \mathrm{C}^{2} \cdot \mathrm{J}^{-1},$ show that the units of permittivity are $\mathrm{F} / \mathrm{m}$. See Appendix $\mathrm{B}$.
Solve the given problems.When studying a solar energy system, the units encountered are $\mathrm{kg} \cdot \mathrm{s}^{-1}\left(\mathrm{m} \cdot \mathrm{s}^{-2}\right)^{2} .$ Simplify these units and include joules (see Exercise 71 ) and only positive exponents in the final result.
Solve the given problems.The metric units for the velocity v of an object are $\mathrm{m} \cdot \mathrm{s}^{-1}$, and the units for the acceleration $a$ of the object are $m \cdot s^{-2} .$ What are the units for $v / a ?$
Solve the given problems.Given that $v=a^{p} t^{r},$ where $v$ is the velocity of an object, $a$ is its acceleration, and $t$ is the time, use the metric units given in Exercise 74 to show that $p=r=1$
Solve the given problems.In optics, the combined focal length $F$ of two lenses is given by $F=\left[f_{1}^{-1}+f_{2}^{-1}+d\left(f_{1} f_{2}\right)^{-1}\right]^{-1},$ where $f_{1}$ and $f_{2}$ are the focal lengths of the lenses and $d$ is the distance between them. Simplify the right side of this cauation.
Solve the given problems.The monthly loan payment $P$ for loan amount $A$ with an annual interest rate $r$ (as a decimal) for $t$ years is $$P=\frac{A\left(\frac{r}{12}\right)}{1-\left(1+\frac{r}{12}\right)^{-12 t}}$$ Find the monthly payment for a 20,000 dollars car loan if it is a 5 -year loan with an annual interest rate of $4 \%$
Solve the given problems.An idealized model of the thermodynamic process in a gasoline engine is the Otto cycle. The efficiency $e$ of the process is $$e=\frac{\frac{T_{1} r^{\gamma}}{r}-\frac{T_{2} r^{\gamma}}{r}-T_{1}+T_{2}}{\frac{T_{1} r^{\gamma}}{r}-\frac{T_{2} r^{\gamma}}{r}} \quad \text { Show that } e=1-\frac{1}{r^{\gamma-1}}$$