Chapter 7

The amount of a certain medicine present in the bloodstream decreases at a rate of 10\(\%\) per hour. a. Which is a better model to use for this scenario: \(A=A_{0}(1+r)^{t}\) or \(A=A_{0} e^{r t} ?\) Explain your answer. b. Using both models, find the amount of medicine in the bloodstream after 10.5 hours if the initial dose was 200 milligrams.

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ f(x)=(2 x)^{-6} \div x^{3} ; f(-3) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 7^{\frac{3}{4}} \times 7^{\frac{5}{4}} $$

Solve each equation and check. \(2^{2 x+1}=16^{x}\)

Show that if the area of one face of a cube is \(B,\) the volume of the cube is \(B^{\frac{3}{2}}\)

Simplify each expression. In each exercise, all variables are positive. \(\frac{4(a b)^{2} c^{5}}{a b c}\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=\left(x^{-7}\right)^{4} ; \mathrm{f}(-6) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 4 \times 4^{\frac{1}{2}} $$

Solve each equation and check. \(9^{x-1}=27^{x}\)

If the area of one face of a cube is \(B\) and the volume of the cube is \(V,\) express \(B\) in terms of \(V\)

Simplify each expression. In each exercise, all variables are positive. \(\frac{\left(a^{x}\right)^{y} b}{a^{x y}}\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ f(x)=\left(\frac{1}{x}+\frac{3}{2}\right)^{-2} ; f(2) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 32 \times 32^{\frac{1}{3}} $$

Solve each equation and check. \(100^{x}=1,000^{x-1}\)

What is the value of \(n\) if \(8^{3}=2^{n} ?\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=10^{x}+10^{-2 x} ; \mathrm{f}(3) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 2^{\frac{1}{4}} \times 8^{1} $$

Solve each equation and check. \(125^{x-1}=25^{x}\)

What is the value of \(a\) if \(27^{2}=9^{a} ?\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ f(x)=x^{-7} \div x^{8} ; f\left(\frac{3}{4}\right) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 12^{\frac{5}{3}} \div 12^{\frac{2}{3}} $$

Solve each equation and check. \(6^{2-x}=\left(\frac{1}{36}\right)^{2}\)

If \(3^{a+1}=x\) and \(3^{a}=y,\) express \(y\) in terms of \(x\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=\left(3 x^{-3}-2 x^{-3}\right)^{2} ; \mathrm{f}(-2) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 3^{\frac{7}{3}} \div 3^{\frac{1}{3}} $$

Solve each equation and check. \(\left(\frac{1}{4}\right)^{x}=8^{1-x}\)

If \(25^{b+1}=x\) and \(5^{b}=y,\) express \(x\) in terms of \(y\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=x^{8}\left(x^{-2}+\frac{1}{x^{3}}\right) ; \mathrm{f}\left(\frac{1}{2}\right) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 4^{\frac{1}{3}} \div 4^{\frac{1}{6}} $$

Solve each equation and check. \(\left(\frac{1}{3}\right)^{x}=9^{1-x}\)

The formula \(A=P(1+r)^{t}\) expresses the amount \(A\) to which \(P\) dollars will increase if invested for \(t\) years at a rate of \(r\) per year. Find \(A\) when \(P=\$ 500, r=0.04\) and \(t=5\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=\left(\frac{x^{-1}}{(2 x)^{-2}}\right)^{-1} ; \mathrm{f}(8) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 125^{\frac{2}{3}} \div 125^{\frac{1}{3}} $$

Solve each equation and check. \((0.01)^{2 x}=100^{2-x}\)

The formula \(A=P(1+r)^{t}\) expresses the amount \(A\) to which \(P\) dollars will increase if invested for \(t\) years at a rate of \(r\) per year. Find the amount to which \(\$ 2,400\) will increase when invested at 5\(\%\) for 10 years.

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ f(x)=\frac{1}{1+\frac{2}{x^{-1}}} ; f(-5) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 4^{0}+4^{-\frac{1}{2}} $$

Solve each equation and check. \((0.25)^{x-2}=4^{x}\)

The formula \(A=P(1+r)^{t}\) expresses the amount \(A\) to which \(P\) dollars will increase if invested for \(t\) years at a rate of \(r\) per year. What is the minimum number of years that \(\$ 1\) must be in invested at 5\(\%\) to increase to \(\$ 2 ?\) (Use a calculator to try possible values of \(t.\))

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=4\left(\frac{1}{2}\right)^{-x}+3\left(\frac{1}{2}\right)^{-x} ; \mathrm{f}(3) $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 9^{-2}+9^{\frac{1}{2}} $$

Solve each equation and check. \(5^{x-1}=(0.04)^{2 x}\)

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ x^{-4} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 2\left[(3)^{-2}+(4)^{-2}\right]^{-\frac{1}{2}} $$

Solve each equation and check. \(4^{x}+7=15\)

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ a^{-6} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ \left(2.3 \times 10^{-1}\right)\left(5.2 \times 10^{-3}\right) $$

Solve each equation and check. \(5+7^{x}=6\)

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ y^{-5} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ \frac{\left(2(3)^{2}+\frac{1}{3^{-2}}\right)^{\frac{2}{3}}}{6\left(2+\frac{1}{4}\right)^{-\frac{1}{2}}} $$