8.4 Add, Subtract, and Multiply Radical Expressions

Before you get started, take this readiness quiz.

Add: $3 x^{2} + 9 x - 5 - \left(x^{2} - 2 x + 3\right) .$
If you missed this problem, review the reference.

Simplify: $\left(2 + a\right) \left(4 - a\right) .$
If you missed this problem, review the reference.

Simplify: $\left(9 - 5 y\right)^{2} .$
If you missed this problem, review the reference.

Simplify: ⓐ $2 \sqrt{2} - 7 \sqrt{2}$ ⓑ $5 \sqrt[3]{y} + 4 \sqrt[3]{y}$ ⓒ $7 \sqrt[4]{x} - 2 \sqrt[4]{y} .$

Simplify: ⓐ $8 \sqrt{2} - 9 \sqrt{2}$ ⓑ $4 \sqrt[3]{x} + 7 \sqrt[3]{x}$ ⓒ $3 \sqrt[4]{x} - 5 \sqrt[4]{y} .$

Simplify: ⓐ $5 \sqrt{3} - 9 \sqrt{3}$ ⓑ $5 \sqrt[3]{y} + 3 \sqrt[3]{y}$ ⓒ $5 \sqrt[4]{m} - 2 \sqrt[3]{m} .$

Simplify: ⓐ $2 \sqrt{5 n} - 6 \sqrt{5 n} + 4 \sqrt{5 n}$ ⓑ $\sqrt[4]{3 x y} + 5 \sqrt[4]{3 x y} - 4 \sqrt[4]{3 x y} .$

Simplify: ⓐ $\sqrt{7 x} - 7 \sqrt{7 x} + 4 \sqrt{7 x}$ ⓑ $4 \sqrt[4]{5 x y} + 2 \sqrt[4]{5 x y} - 7 \sqrt[4]{5 x y} .$

Simplify: ⓐ $4 \sqrt{3 y} - 7 \sqrt{3 y} + 2 \sqrt{3 y}$ ⓑ $6 \sqrt[3]{7 m n} + \sqrt[3]{7 m n} - 4 \sqrt[3]{7 m n} .$

Simplify: ⓐ $\sqrt{20} + 3 \sqrt{5}$ ⓑ $\sqrt[3]{24} - \sqrt[3]{375}$ ⓒ $\frac{1}{2} \sqrt[4]{48} - \frac{2}{3} \sqrt[4]{243} .$

Simplify: ⓐ $\sqrt{18} + 6 \sqrt{2}$ ⓑ $6 \sqrt[3]{16} - 2 \sqrt[3]{250}$ ⓒ $\frac{2}{3} \sqrt[3]{81} - \frac{1}{2} \sqrt[3]{24} .$

Simplify: ⓐ $\sqrt{27} + 4 \sqrt{3}$ ⓑ $4 \sqrt[3]{5} - 7 \sqrt[3]{40}$ ⓒ $\frac{1}{2} \sqrt[3]{128} - \frac{5}{3} \sqrt[3]{54} .$

Simplify: ⓐ $9 \sqrt{50 m^{2}} - 6 \sqrt{48 m^{2}}$ ⓑ $\sqrt[3]{54 n^{5}} - \sqrt[3]{16 n^{5}} .$

Simplify: ⓐ $\sqrt{32 m^{7}} - \sqrt{50 m^{7}}$ ⓑ $\sqrt[3]{135 x^{7}} - \sqrt[3]{40 x^{7}} .$

Simplify: ⓐ $\sqrt{27 p^{3}} - \sqrt{48 p^{3}}$ ⓑ $\sqrt[3]{256 y^{5}} - \sqrt[3]{32 y^{5}} .$

Simplify: ⓐ $\left(6 \sqrt{2}\right) \left(3 \sqrt{10}\right)$ ⓑ $\left(−5 \sqrt[3]{4}\right) \left(−4 \sqrt[3]{6}\right) .$

Simplify: ⓐ $\left(3 \sqrt{2}\right) \left(2 \sqrt{30}\right)$ ⓑ $\left(2 \sqrt[3]{18}\right) \left(−3 \sqrt[3]{6}\right) .$

Simplify: ⓐ $\left(3 \sqrt{3}\right) \left(3 \sqrt{6}\right)$ ⓑ $\left(−4 \sqrt[3]{9}\right) \left(3 \sqrt[3]{6}\right) .$

Simplify: ⓐ $\left(10 \sqrt{6 p^{3}}\right) \left(4 \sqrt{3 p}\right)$ ⓑ $\left(2 \sqrt[4]{20 y^{2}}\right) \left(3 \sqrt[4]{28 y^{3}}\right) .$

Simplify: ⓐ $\left(6 \sqrt{6 x^{2}}\right) \left(8 \sqrt{30 x^{4}}\right)$ ⓑ $\left(−4 \sqrt[4]{12 y^{3}}\right) \left(− \sqrt[4]{8 y^{3}}\right) .$

Simplify: ⓐ $\left(2 \sqrt{6 y^{4}}\right) \left(12 \sqrt{30 y}\right)$ ⓑ $\left(−4 \sqrt[4]{9 a^{3}}\right) \left(3 \sqrt[4]{27 a^{2}}\right) .$

Simplify: ⓐ $\sqrt{6 } \left(\sqrt{2} + \sqrt{18}\right)$ ⓑ $\sqrt[3]{9} \left(5 - \sqrt[3]{18}\right) .$

Simplify: ⓐ $\sqrt{6} \left(1 + 3 \sqrt{6}\right)$ ⓑ $\sqrt[3]{4} \left(−2 - \sqrt[3]{6}\right) .$

Simplify: ⓐ $\sqrt{8} \left(2 - 5 \sqrt{8}\right)$ ⓑ $\sqrt[3]{3} \left(− \sqrt[3]{9} - \sqrt[3]{6}\right) .$

Simplify: ⓐ $\left(3 - 2 \sqrt{7}\right) \left(4 - 2 \sqrt{7}\right)$ ⓑ $\left(\sqrt[3]{x} - 2\right) \left(\sqrt[3]{x} + 4\right) .$

Simplify: ⓐ $\left(6 - 3 \sqrt{7}\right) \left(3 + 4 \sqrt{7}\right)$ ⓑ $\left(\sqrt[3]{x} - 2\right) \left(\sqrt[3]{x} - 3\right) .$

Simplify: ⓐ $\left(2 - 3 \sqrt{11}\right) \left(4 - \sqrt{11}\right)$ ⓑ $\left(\sqrt[3]{x} + 1\right) \left(\sqrt[3]{x} + 3\right) .$

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

Simplify: $\left(5 \sqrt{3} - \sqrt{7}\right) \left(\sqrt{3} + 2 \sqrt{7}\right)$

Simplify: $\left(\sqrt{6} - 3 \sqrt{8}\right) \left(2 \sqrt{6} + \sqrt{8}\right)$

Simplify: ⓐ $\left(10 + \sqrt{2}\right)^{2}$ ⓑ $\left(1 + 3 \sqrt{6}\right)^{2} .$

Simplify: ⓐ $\left(6 - \sqrt{5}\right)^{2}$ ⓑ $\left(9 - 2 \sqrt{10}\right)^{2} .$

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

Simplify: $\left(4 + 5 \sqrt{7}\right) \left(4 - 5 \sqrt{7}\right) .$

Simplify.

ⓐ $8 \sqrt{2} - 5 \sqrt{2}$ ⓑ $5 \sqrt[3]{m} + 2 \sqrt[3]{m}$ ⓒ $8 \sqrt[4]{m} - 2 \sqrt[4]{m}$

Simplify.

ⓐ $3 \sqrt{5} + 6 \sqrt{5}$ ⓑ $9 \sqrt[3]{a} + 3 \sqrt[3]{a}$ ⓒ $5 \sqrt[4]{2 z} + \sqrt[4]{2 z}$

Simplify.

ⓐ $3 \sqrt{2 a} - 4 \sqrt{2 a} + 5 \sqrt{2 a}$ ⓑ $5 \sqrt[4]{3 a b} - 3 \sqrt[4]{3 a b} - 2 \sqrt[4]{3 a b}$

Simplify.

ⓐ $8 \sqrt{3 c} + 2 \sqrt{3 c} - 9 \sqrt{3 c}$ ⓑ $2 \sqrt[3]{4 p q} - 5 \sqrt[3]{4 p q} + 4 \sqrt[3]{4 p q}$

Simplify.

ⓐ $\sqrt{27} - \sqrt{75}$ ⓑ $\sqrt[3]{40} - \sqrt[3]{320}$ ⓒ $\frac{1}{2} \sqrt[4]{32} + \frac{2}{3} \sqrt[4]{162}$

Simplify.

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

Simplify.

ⓐ $\sqrt{72 a^{5}} - \sqrt{50 a^{5}}$ ⓑ $9 \sqrt[4]{80 p^{4}} - 6 \sqrt[4]{405 p^{4}}$

Simplify.

ⓐ $\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$ ⓑ $2 \sqrt[4]{162 r^{10}} + 4 \sqrt[4]{32 r^{10}}$

Simplify.

$3 \sqrt{128 y^{2}} + 4 y \sqrt{162} - 8 \sqrt{98 y^{2}}$

Simplify.

ⓐ $\left(−2 \sqrt[]{3}\right) \left(3 \sqrt[]{18}\right)$ ⓑ $\left(8 \sqrt[3]{4}\right) \left(−4 \sqrt[3]{18}\right)$

Simplify.

ⓐ $\left(5 \sqrt[]{6}\right) \left(− \sqrt[]{12}\right)$ ⓑ $\left(−2 \sqrt[4]{18}\right) \left(− \sqrt[4]{9}\right)$

Simplify.

ⓐ $\left(4 \sqrt{12 z^{3}}\right) \left(3 \sqrt{9 z}\right)$ ⓑ $\left(5 \sqrt[3]{3 x^{3}}\right) \left(3 \sqrt[3]{18 x^{3}}\right)$

Simplify.

ⓐ $\left(−2 \sqrt{7 z^{3}}\right) \left(3 \sqrt{14 z^{8}}\right)$ ⓑ $\left(2 \sqrt[4]{8 y^{2}}\right) \left(−2 \sqrt[4]{12 y^{3}}\right)$

Multiply.

ⓐ $\sqrt{7} \left(5 + 2 \sqrt{7}\right)$ ⓑ $\sqrt[3]{6} \left(4 + \sqrt[3]{18}\right)$

Multiply.

ⓐ $\sqrt{11} \left(−3 + 4 \sqrt{11}\right)$ ⓑ $\sqrt[4]{3} \left(\sqrt[4]{54} + \sqrt[4]{18}\right)$

Multiply.

$\left(7 + \sqrt{3}\right) \left(9 - \sqrt{3}\right)$

Multiply.

ⓐ $\left(9 - 3 \sqrt{2}\right) \left(6 + 4 \sqrt{2}\right)$ ⓑ $\left(\sqrt[3]{x} - 3\right) \left(\sqrt[3]{x} + 1\right)$

Multiply.

ⓐ $\left(1 + 3 \sqrt{10}\right) \left(5 - 2 \sqrt{10}\right)$ ⓑ $\left(2 \sqrt[3]{x} + 6\right) \left(\sqrt[3]{x} + 1\right)$

Multiply.

$\left(\sqrt{3} + \sqrt{10}\right) \left(\sqrt{3} + 2 \sqrt{10}\right)$

Multiply.

$\left(2 \sqrt{7} - 5 \sqrt{11}\right) \left(4 \sqrt{7} + 9 \sqrt{11}\right)$

Multiply.

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

Multiply.

ⓐ $\left(9 - \sqrt{6}\right)^{2}$ ⓑ $\left(10 + 3 \sqrt{7}\right)^{2}$

Multiply.

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

Multiply.

$\left(4 + 9 \sqrt{3}\right) \left(4 - 9 \sqrt{3}\right)$

Multiply.

$\left(12 - 5 \sqrt{5}\right) \left(12 + 5 \sqrt{5}\right)$

Multiply.

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

Mixed Practice

$\frac{2}{3} \sqrt{27} + \frac{3}{4} \sqrt{48}$

Mixed Practice

$\frac{5}{6} \sqrt{162} + \frac{3}{16} \sqrt{128}$

Mixed Practice

$\frac{1}{2} \sqrt[4]{80} - \frac{2}{3} \sqrt[4]{405}$

Mixed Practice

$5 \sqrt{12 c^{4}} - 3 \sqrt{27 c^{6}}$

Mixed Practice

$\frac{3}{5} \sqrt{75} - \frac{1}{4} \sqrt{48}$

Mixed Practice

$8 \sqrt[3]{64 q^{6}} - 3 \sqrt[3]{125 q^{6}}$

Mixed Practice

$\sqrt{3} \cdot \sqrt{21}$

Mixed Practice

$\left(7 \sqrt[3]{4}\right) \left(−3 \sqrt[3]{18}\right)$

Mixed Practice

$\left(\sqrt{29}\right)^{2}$

Mixed Practice

$\left(−4 + \sqrt{17}\right) \left(−3 + \sqrt{17}\right)$

Mixed Practice

$\left(6 - 3 \sqrt{2}\right)^{2}$

Mixed Practice

$\sqrt[3]{3} \left(2 \sqrt[3]{9} + \sqrt[3]{18}\right)$

Explain when a radical expression is in simplest form.

ⓐ Explain why $\left(− \sqrt{n}\right)^{2}$ is always non-negative, for $n \geq 0 .$
ⓑ Explain why $- \left(\sqrt{n}\right)^{2}$ is always non-positive, for $n \geq 0.$