Chapter 7

Show that the formula \(A=A_{0}(1+r)^{n}\) is equivalent to \(A=A_{0}(2)^{n}\) when \(r=100 \%\)

Kim said that \(a^{0}+a^{0}=a^{0+0}=a^{0}=1 .\) Do you agree with Kim? Explain why or why not.

Use exponents to show that for \(a>0,(\sqrt[n]{a})^{0}=1\)

Ethan said that to solve the equation \((x+3)^{\frac{1}{2}}=5,\) the first step should be to square both sides of the equation. Do you agree with Ethan? Explain why or why not.

What value of \(a\) makes the equation \(6^{a}=1\) true? Justify your answer.

Randy said that \((2)^{3}(5)^{2}=(10)^{5} .\) Do you agree with Randy? Justify your answer.

Explain why, if an investment is earning interest at a rate of 5\(\%\) per year, the investment is worth more if the interest is compounded daily rather than if it is compounded yearly.

Tony said that \(a^{0}+a^{0}=2 a^{0}=2 .\) Do you agree with Tony? Explain why or why not?

Use exponents to show that for \(a>0, \sqrt{\sqrt{a}}=\sqrt[4]{a}\)

Chloe changed the equation \(a^{-2}=36\) to the equation \(\frac{1}{a^{2}}=\frac{1}{36}\) and then took the square root of each side. Will Chloe's solution be correct? Explain why or why not.

Explain why the equation \(3^{a}=5^{a-1}\) cannot be solved using the procedure used in this section.

Explain why \(y=b^{x}\) is not an exponential function for \(b=1\)

Natasha said that \((2)^{3}(5)^{3}=(10)^{3} .\) Do you agree with Natasha? Justify your answer.

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{2} $$

In \(3-10,\) write each expression as a rational number without an exponent. $$ 5^{-1} $$

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

In \(3-17\) solve each equation and check. $$ x^{\frac{1}{3}}=4 $$

Write each number as a power. 9

In \(3-6 :\) a. Sketch the graph of each function. b. On the same set of axes, sketch the graph of the image of the reflection in the \(y\) -axis of the graph drawn in part a. . Write an equation of the graph of the function drawn in part b. $$ y=4^{x} $$

Simplify each expression. In each exercise, all variables are positive. \(x^{3} \cdot x^{4}\)

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{1.5} $$

In \(3-10,\) write each expression as a rational number without an exponent. $$ 4^{-2} $$

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

Write each number as a power. 27

In \(3-17\) solve each equation and check. $$ a^{\frac{1}{5}}=2 $$

In \(3-6 :\) a. Sketch the graph of each function. b. On the same set of axes, sketch the graph of the image of the reflection in the \(y\) -axis of the graph drawn in part a. . Write an equation of the graph of the function drawn in part b. $$ y=3^{x} $$

Simplify each expression. In each exercise, all variables are positive. \(y \cdot y^{5}\)

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{-1} $$

In \(3-10,\) write each expression as a rational number without an exponent. $$ 6^{-2} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 100^{\frac{1}{2}} $$

Write each number as a power. 25

In \(3-17\) solve each equation and check. $$ x^{\frac{2}{3}}=9 $$

In \(3-6 :\) a. Sketch the graph of each function. b. On the same set of axes, sketch the graph of the image of the reflection in the \(y\) -axis of the graph drawn in part a. . Write an equation of the graph of the function drawn in part b. $$ y=\left(\frac{7}{2}\right)^{x} $$

Simplify each expression. In each exercise, all variables are positive. \(x^{6} \div x^{2}\)

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x e^{3}=e^{4} $$

In \(3-10,\) write each expression as a rational number without an exponent. $$ \left(\frac{1}{2}\right)^{-1} $$

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

Write each number as a power. 49

In \(3-17\) solve each equation and check. $$ b^{\frac{3}{2}}=8 $$

In \(3-6 :\) a. Sketch the graph of each function. b. On the same set of axes, sketch the graph of the image of the reflection in the \(y\) -axis of the graph drawn in part a. . Write an equation of the graph of the function drawn in part b. $$ y=\left(\frac{3}{4}\right)^{x} $$

Simplify each expression. In each exercise, all variables are positive. \(y^{4} \div y\)

In \(3-10,\) write each expression as a rational number without an exponent. $$ \left(\frac{1}{5}\right)^{-3} $$

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

Write each number as a power. \(1,000\)

In \(3-17\) solve each equation and check. $$ x^{-2}=9 $$

a. Sketch the graph of \(y=\left(\frac{5}{3}\right)^{x}\) b. From the graph of \(y=\left(\frac{5}{3}\right)^{x},\) estimate the value of \(y,\) to the nearest tenth, when \(x=2.2\) c. From the graph of \(y=\left(\frac{5}{3}\right)^{x},\) estimate the value of \(x,\) to the nearest tenth, when \(y=2.9\)

Simplify each expression. In each exercise, all variables are positive. \(\left(x^{5}\right)^{2}\)

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ \frac{x}{e^{3}}=e^{-2} $$

In \(3-10,\) write each expression as a rational number without an exponent. $$ \left(\frac{2}{3}\right)^{-1} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 216^{\frac{1}{3}} $$