Chapter 6

Solve each equation. $$x^{\frac{2}{5}}=2$$

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{3 i}{2+4 i}$$

Solve each equation. $$(2 x+6)^{\frac{1}{2}}=x$$

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{4 i}{5+2 i}$$

Solve each equation. $$(2 x-4)^{\frac{2}{3}}=1$$

A 24-foot ladder resting against a house reaches a windowsill 16 feet above the ground. How far is the foot of the ladder from the foundation of the house? Express your answer to the nearest tenth of a foot.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-2 i}{3-5 i}$$

Solve each equation. $$(4 x+5)^{\frac{2}{3}}=2$$

A 62-foot guy-wire makes an angle of \(60^{\circ}\) with the ground and is attached to a telephone pole (see Figure 6.6). Find the distance from the base of the pole to the point on the pole where the wire is attached. Express your answer to the nearest tenth of a foot.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-5 i}{2-4 i}$$

Solve each equation. $$(6 x+7)^{\frac{1}{2}}=x+2$$

A rectangular plot measures 16 meters by 34 meters. Find, to the nearest meter, the distance from one corner of the plot to the corner diagonally opposite.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-2+6 i}{3 i}$$

Solve each equation. $$(5 x+21)^{\frac{1}{2}}=x+3$$

Consecutive bases of a square-shaped baseball diamond are 90 feet apart (see Figure 6.7). Find, to the nearest tenth of a foot, the distance from first base diagonally across the diamond to third base.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-4-7 i}{6 i}$$

A diagonal of a square parking lot is 75 meters. Find, to the nearest meter, the length of a side of the lot.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{2}{7 i}$$

Explain why the equation \((x+2)^{2}+5=1\) has no real number solutions.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{3}{10 i}$$

Suppose that your friend solved the equation \((x+3)^{2}=\) 25 as follows: $$ \begin{aligned} (x+3)^{2} &=25 \\ x^{2}+6 x+9 &=25 \\ x^{2}+6 x-16 &=0 \end{aligned} $$ $$ \begin{aligned} (x+8)(x-2) &=0 & & & \\ x+8 &=0 \quad \text { or } & & x-2 &=0 \\ x &=-8 & \text { or } & x &=2 \end{aligned} $$ Is this a correct approach to the problem? Would you offer any suggestion about an easier approach to the problem?

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{2+6 i}{1+7 i}$$

Suppose that we are given a cube with edges 12 centimeters in length. Find the length of a diagonal from a lower corner to the diagonally opposite upper corner. Express your answer to the nearest tenth of a centimeter.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{5+i}{2+9 i}$$

Suppose that we are given a rectangular box with a length of 8 centimeters, a width of 6 centimeters, and a height of 4 centimeters. Find the length of a diagonal from a lower corner to the upper corner diagonally opposite. Express your answer to the nearest tenth of a centimeter.

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{3+6 i}{4-5 i}$$

The converse of the Pythagorean theorem is also true. It states, "If the measures \(a, b\), and \(c\) of the sides of a triangle are such that \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle with \(a\) and \(b\) the measures of the legs and \(c\) the measure of the hypotenuse." Use the converse of the Pythagorean theorem to determine which of the triangles with sides of the following measures are right triangles. (a) \(9,40,41\) (b) \(20,48,52\) (c) \(19,21,26\) (d) \(32,37,49\) (e) \(65,156,169\) (f) \(21,72,75\)

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{7-3 i}{4-3 i}$$

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-2+7 i}{-1+i}$$

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-3+8 i}{-2+i}$$

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-1-3 i}{-2-10 i}$$

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-3-4 i}{-4-11 i}$$

Some of the solution sets for quadratic equations in the next sections will contain complex numbers such as \((-4+\sqrt{-12}) / 2\) and \((-4-\sqrt{-12}) / 2\). We can simplify the first number as follows. $$ \begin{array}{r} \frac{-4+\sqrt{-12}}{2}=\frac{-4+i \sqrt{12}}{2}= \\ \frac{-4+2 i \sqrt{3}}{2}=\frac{2(-2+i \sqrt{3})}{2}=-2+i \sqrt{3} \end{array} $$ Simplify each of the following complex numbers. (a) \(\frac{-4-\sqrt{-12}}{2}\) (b) \(\frac{6+\sqrt{-24}}{4}\) (c) \(\frac{-1-\sqrt{-18}}{2}\) (d) \(\frac{-6+\sqrt{-27}}{3}\) (e) \(\frac{10+\sqrt{-45}}{4}\) (f) \(\frac{4-\sqrt{-48}}{2}\)

Why is the set of real numbers a subset of the set of complex numbers?

Can the sum of two nonreal complex numbers be a real number? Defend your answer.

Can the product of two nonreal complex numbers be a real number? Defend your answer.