8.8 Use the Complex Number System

Before you get started, take this readiness quiz.

Given the numbers $−4 , - \sqrt{7} , 0. \overset{–}{5} , \frac{7}{3} , 3 , \sqrt{81} ,$ list the ⓐ rational numbers, ⓑ irrational numbers, ⓒ real numbers.
If you missed this problem, review the reference.

Multiply: $\left(x - 3\right) \left(2 x + 5\right) .$
If you missed this problem, review the reference.

Rationalize the denominator:$\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}} .$
If you missed this problem, review the reference.

Write each expression in terms of i and simplify if possible:

ⓐ $\sqrt{−25}$ ⓑ $\sqrt{−7}$ ⓒ $\sqrt{−12} .$

Write each expression in terms of i and simplify if possible:

ⓐ $\sqrt{−81}$ ⓑ $\sqrt{−5}$ ⓒ $\sqrt{−18} .$

Write each expression in terms of i and simplify if possible:

ⓐ $\sqrt{−36}$ ⓑ $\sqrt{−3}$ ⓒ $\sqrt{−27} .$

Add: $\sqrt{−12} + \sqrt{−27} .$

Add: $\sqrt{−8} + \sqrt{−32} .$

Add: $\sqrt{−27} + \sqrt{−48} .$

Simplify: ⓐ $\left(4 - 3 i\right) + \left(5 + 6 i\right)$ ⓑ $\left(2 - 5 i\right) - \left(5 - 2 i\right) .$

Simplify: ⓐ $\left(2 + 7 i\right) + \left(4 - 2 i\right)$ ⓑ $\left(8 - 4 i\right) - \left(2 - i\right) .$

Simplify: ⓐ $\left(3 - 2 i\right) + \left(−5 - 4 i\right)$ ⓑ $\left(4 + 3 i\right) - \left(2 - 6 i\right) .$

Multiply: $2 i \left(7 - 5 i\right) .$

Multiply: $4 i \left(5 - 3 i\right) .$

Multiply: $−3 i \left(2 + 4 i\right) .$

Multiply: $\left(3 + 2 i\right) \left(4 - 3 i\right) .$

Multiply: $\left(5 - 3 i\right) \left(−1 - 2 i\right) .$

Multiply: $\left(−4 - 3 i\right) \left(2 + i\right) .$

Multiply using the Binomial Squares pattern: $\left(−2 - 5 i\right)^{2} .$

Multiply using the Binomial Squares pattern: $\left(−5 + 4 i\right)^{2} .$

Multiply: $\sqrt{−36} \cdot \sqrt{−4} .$

Multiply: $\sqrt{−49} \cdot \sqrt{−4} .$

Multiply: $\sqrt{−36} \cdot \sqrt{−81} .$

Multiply: $\left(3 - \sqrt{−12}\right) \left(5 + \sqrt{−27}\right) .$

Multiply: $\left(4 - \sqrt{−12}\right) \left(3 - \sqrt{−48}\right) .$

Multiply: $\left(−2 + \sqrt{−8}\right) \left(3 - \sqrt{−18}\right) .$

Multiply: $\left(3 - 2 i\right) \left(3 + 2 i\right) .$

Multiply: $\left(4 - 3 i\right) \cdot \left(4 + 3 i\right) .$

Multiply: $\left(−2 + 5 i\right) \cdot \left(−2 - 5 i\right) .$

Multiply using the Product of Complex Conjugates Pattern: $\left(3 - 10 i\right) \left(3 + 10 i\right) .$

Multiply using the Product of Complex Conjugates Pattern: $\left(−5 + 4 i\right) \left(−5 - 4 i\right) .$

Divide: $\frac{2 + 5 i}{5 - 2 i} .$

Divide: $\frac{1 + 6 i}{6 - i} .$

Divide, writing the answer in standard form: $\frac{−3}{5 + 2 i} .$

Divide, writing the answer in standard form: $\frac{4}{1 - 4 i} .$

Divide, writing the answer in standard form: $\frac{−2}{−1 + 2 i} .$

Divide: $\frac{5 + 3 i}{4 i} .$

Divide: $\frac{3 + 3 i}{2 i} .$

Divide: $\frac{2 + 4 i}{5 i} .$

Simplify: $i^{75} .$

Simplify: $i^{92} .$

Write each expression in terms of i and simplify if possible.

ⓐ $\sqrt{−16}$ ⓑ $\sqrt{−11}$
ⓒ $\sqrt{−8}$

Write each expression in terms of i and simplify if possible.

ⓐ $\sqrt{−100}$ ⓑ $\sqrt{−13}$ ⓒ $\sqrt{−45}$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\sqrt{−75} + \sqrt{−48}$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\sqrt{−50} + \sqrt{−18}$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\left(1 + 3 i\right) + \left(7 + 4 i\right)$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\left(8 - i\right) + \left(6 + 3 i\right)$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\left(1 - 4 i\right) - \left(3 - 6 i\right)$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\left(6 + i\right) - \left(−2 - 4 i\right)$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\left(5 - \sqrt{−36}\right) + \left(2 - \sqrt{−49}\right)$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\left(−7 - \sqrt{−50}\right) - \left(−32 - \sqrt{−18}\right)$

Multiply.

$4 i \left(5 - 3 i\right)$

Multiply.

$−6 i \left(−3 - 2 i\right)$

Multiply.

$\left(4 + 3 i\right) \left(−5 + 6 i\right)$

Multiply.

$\left(−3 + 3 i\right) \left(−2 - 7 i\right)$

Multiply using the Product of Binomial Squares Pattern.

$\left(3 + 4 i\right)^{2}$

Multiply using the Product of Binomial Squares Pattern.

$\left(−2 - 3 i\right)^{2}$

Multiply.

$\sqrt{−25} \cdot \sqrt{−36}$

Multiply.

$\sqrt{−9} \cdot \sqrt{−100}$

Multiply.

$\left(−2 - \sqrt{−27}\right) \left(4 - \sqrt{−48}\right)$

Multiply.

$\left(2 + \sqrt{−8}\right) \left(−4 + \sqrt{−18}\right)$

Multiply.

$\left(2 - i\right) \left(2 + i\right)$

Multiply.

$\left(7 - 2 i\right) \left(7 + 2 i\right)$

Multiply using the Product of Complex Conjugates Pattern.

$\left(7 - i\right) \left(7 + i\right)$

Multiply using the Product of Complex Conjugates Pattern.

$\left(9 - 2 i\right) \left(9 + 2 i\right)$

Divide.

$\frac{3 + 4 i}{4 - 3 i}$

Divide.

$\frac{2 + i}{3 - 4 i}$

Divide.

$\frac{3}{2 - 3 i}$

Divide.

$\frac{−4}{3 - 2 i}$

Divide.

$\frac{1 + 4 i}{3 i}$

Divide.

$\frac{−2 - 3 i}{4 i}$

Simplify.

$i^{41}$

Simplify.

$i^{66}$

Simplify.

$i^{128}$

Simplify.

$i^{137}$

Explain the relationship between real numbers and complex numbers.

Why is $\sqrt{−64} = 8 i$ but $\sqrt[3]{−64} = −4 .$

Simplify.

ⓐ $\sqrt{225}$ ⓑ $− \sqrt{16}$

Simplify.

ⓐ $\sqrt[3]{8}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{243}$

Estimate each root between two consecutive whole numbers.

ⓐ $\sqrt{68}$ ⓑ $\sqrt[3]{84}$

Simplify using absolute values as necessary.

ⓐ $\sqrt[3]{a^{3}}$
ⓑ $\sqrt[7]{b^{7}}$

Simplify using absolute values as necessary.

ⓐ $\sqrt[4]{m^{8}}$
ⓑ $\sqrt[5]{n^{20}}$

Simplify using absolute values as necessary.

ⓐ $\sqrt[3]{216 a^{6}}$
ⓑ $\sqrt[5]{32 b^{20}}$

Use the Product Property to simplify radical expressions.

$\sqrt{125}$

Use the Product Property to simplify radical expressions.

ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt{80 s^{15}}$
ⓑ $\sqrt[5]{96 a^{7}}$
ⓒ $\sqrt[6]{128 b^{7}}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt[5]{−32}$
ⓑ $\sqrt[8]{−1}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{72}{98}}$ ⓑ $\sqrt[3]{\frac{24}{81}}$ ⓒ $\sqrt[4]{\frac{6}{96}}$

Use the Quotient Property to simplify square roots.

$\sqrt{\frac{300 m^{5}}{64}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$
ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$
ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

Write as a radical expression.

ⓐ $r^{\frac{1}{2}}$ ⓑ $s^{\frac{1}{3}}$ ⓒ $t^{\frac{1}{4}}$

Simplify.

ⓐ $625^{\frac{1}{4}}$
ⓑ $243^{\frac{1}{5}}$
ⓒ $32^{\frac{1}{5}}$

Simplify.

ⓐ $\left(−32\right)^{\frac{1}{5}}$
ⓑ $\left(243\right)^{- \frac{1}{5}}$
ⓒ $− 125^{\frac{1}{3}}$

Simplify.

ⓐ $25^{\frac{3}{2}}$
ⓑ $9^{- \frac{3}{2}}$
ⓒ $\left(−64\right)^{\frac{2}{3}}$

Simplify.

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$
ⓑ $\left(b^{15}\right)^{\frac{3}{5}}$
ⓒ $\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

Simplify.

ⓐ $7 \sqrt{2} - 3 \sqrt{2}$
ⓑ $7 \sqrt[3]{p} + 2 \sqrt[3]{p}$
ⓒ $5 \sqrt[3]{x} - 3 \sqrt[3]{x}$

Simplify.

ⓐ $\sqrt{48} + \sqrt{27}$
ⓑ $\sqrt[3]{54} + \sqrt[3]{128}$
ⓒ $6 \sqrt[4]{5} - \frac{3}{2} \sqrt[4]{80}$

Simplify.

$3 \sqrt{75 y^{2}} + 8 y \sqrt{48} - \sqrt{300 y^{2}}$

Simplify.

ⓐ $\left(3 \sqrt{2 x^{3}}\right) \left(7 \sqrt{18 x^{2}}\right)$
ⓑ $\left(−6 \sqrt[3]{20 a^{2}}\right) \left(−2 \sqrt[3]{16 a^{3}}\right)$

Multiply.

ⓐ $\left(3 - 2 \sqrt{7}\right) \left(5 - 4 \sqrt{7}\right)$
ⓑ $\left(\sqrt[3]{x} - 5\right) \left(\sqrt[3]{x} - 3\right)$

Multiply.

ⓐ $\left(4 + \sqrt{11}\right)^{2}$
ⓑ $\left(3 - 2 \sqrt{5}\right)^{2}$

Multiply.

$\left(\sqrt[3]{3 x} + 2\right) \left(\sqrt[3]{3 x} - 2\right)$

Simplify.

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$
ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

Rationalize the denominator.

ⓐ $\frac{1}{\sqrt[3]{11}}$ ⓑ $\sqrt[3]{\frac{7}{54}}$ ⓒ $\frac{3}{\sqrt[3]{3 x^{2}}}$

Simplify.

$\frac{7}{2 - \sqrt{6}}$

Simplify.

$\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}$

Solve.

$\sqrt{5 x + 1} = −3$

Solve.

$\sqrt{u - 3} + 3 = u$

Solve.

$\left(8 x + 5\right)^{\frac{1}{3}} + 2 = −1$

Solve.

$2 \sqrt{8 r + 1} - 8 = 2$

Solve.

$\sqrt[3]{2 x^{2} + 9 x - 18} = \sqrt[3]{x^{2} + 3 x - 2}$

Solve.

$\sqrt{x + 1} - \sqrt{x - 2} = 1$

Solve. Round approximations to one decimal place.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Evaluate each function.

$G \left(x\right) = \sqrt{5 x - 1} ,$ find
ⓐ $G \left(5\right)$
ⓑ $G \left(2\right)$

Evaluate each function.

For the function
$g \left(x\right) = \sqrt[4]{4 - 4 x} ,$ find
ⓐ $g \left(1\right)$
ⓑ $g \left(−3\right)$

Find the domain of the function and write the domain in interval notation.

$F \left(x\right) = \sqrt{\frac{x + 3}{x - 2}}$

Find the domain of the function and write the domain in interval notation.

$F \left(x\right) = \sqrt[4]{10 - 7 x}$

Add or subtract.

$\sqrt{−50} + \sqrt{−18}$

Add or subtract.

$\left(6 + i\right) - \left(−2 - 4 i\right)$

Multiply.

$\left(−2 - 5 i\right) \left(−4 + 3 i\right)$

Multiply.

$\sqrt{−4} \cdot \sqrt{−16}$

Multiply using the Product of Binomial Squares Pattern.

$\left(−2 - 3 i\right)^{2}$

Divide.

$\frac{2 + i}{3 - 4 i}$

Simplify.

$i^{48}$

Simplify using absolute values as necessary.

$\sqrt[3]{125 x^{9}}$

Simplify using absolute values as necessary.

$\sqrt[3]{72 x^{8} y^{4}}$

Simplify. Assume all variables are positive.

ⓐ $256^{- \frac{1}{4}}$ ⓑ $− 49^{\frac{3}{2}}$

Simplify. Assume all variables are positive.

$\frac{x^{- \frac{1}{4}} \cdot x^{\frac{5}{4}}}{x^{- \frac{3}{4}}}$

Simplify. Assume all variables are positive.

$\sqrt{48 x^{5}} - \sqrt{75 x^{5}}$

Simplify. Assume all variables are positive.

$2 \sqrt{12 x^{5}} \cdot 3 \sqrt{6 x^{3}}$

Simplify. Assume all variables are positive.

$\left(4 - 3 \sqrt{3}\right) \left(5 + 2 \sqrt{3}\right)$

Simplify. Assume all variables are positive.

$\frac{\sqrt{245 x y^{−4}}}{\sqrt{45 x^{−4} y^{3}}}$

Simplify. Assume all variables are positive.

$\frac{3}{2 + \sqrt{3}}$

Simplify. Assume all variables are positive.

$−4 i \left(−2 - 3 i\right)$

Simplify. Assume all variables are positive.

$i^{172}$

Solve.

$\sqrt{x + 5} + 1 = x$