9.4 Multiply Square Roots

Before you get started, take this readiness quiz.

Simplify: $\left(3 u\right) \left(8 v\right)$.
If you missed this problem, review the reference.

Simplify: $6 \left(12 - 7 n\right)$.
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: ⓐ $\sqrt{2} \cdot \sqrt{6}$ ⓑ $\left(4 \sqrt{3}\right) \left(2 \sqrt{12}\right)$.

Simplify: ⓐ $\sqrt{3} \cdot \sqrt{6}$ ⓑ $\left(2 \sqrt{6}\right) \left(3 \sqrt{12}\right)$.

Simplify: ⓐ $\sqrt{5} \cdot \sqrt{10}$ ⓑ $\left(6 \sqrt{3}\right) \left(5 \sqrt{6}\right)$.

Simplify: $\left(6 \sqrt{2}\right) \left(3 \sqrt{10}\right)$.

Simplify: $\left(3 \sqrt{2}\right) \left(2 \sqrt{30}\right)$.

Simplify: $\left(3 \sqrt{3}\right) \left(3 \sqrt{6}\right)$.

Simplify: ⓐ $\left(\sqrt{8 x^{3}}\right) \left(\sqrt{3 x}\right)$ ⓑ $\left(\sqrt{20 y^{2}}\right) \left(\sqrt{5 y^{3}}\right)$.

Simplify: ⓐ $\left(\sqrt{6 x^{3}}\right) \left(\sqrt{3 x}\right)$ ⓑ $\left(\sqrt{2 y^{3}}\right) \left(\sqrt{50 y^{2}}\right)$.

Simplify: ⓐ $\left(\sqrt{6 x^{5}}\right) \left(\sqrt{2 x}\right)$ ⓑ $\left(\sqrt{12 y^{2}}\right) \left(\sqrt{3 y^{5}}\right)$.

Simplify: $\left(10 \sqrt{6 p^{3}}\right) \left(3 \sqrt{18 p}\right)$.

Simplify: $\left(6 \sqrt{2 x^{2}}\right) \left(8 \sqrt{45 x^{4}}\right)$.

Simplify: $\left(2 \sqrt{6 y^{4}}\right) \left(12 \sqrt{30 y}\right)$.

Simplify: ⓐ $\left(\sqrt{2}\right)^{2}$ ⓑ $\left(− \sqrt{11}\right)^{2}$.

Simplify: ⓐ $\left(\sqrt{12}\right)^{2}$ ⓑ $\left(− \sqrt{15}\right)^{2}$.

Simplify: ⓐ $\left(\sqrt{16}\right)^{2}$ ⓑ $\left(− \sqrt{20}\right)^{2}$.

Simplify: ⓐ $\left(2 \sqrt{3}\right) \left(8 \sqrt{3}\right)$ ⓑ $\left(3 \sqrt{6}\right)^{2}$.

Simplify: ⓐ $\left(6 \sqrt{11}\right) \left(5 \sqrt{11}\right)$ ⓑ $\left(5 \sqrt{8}\right)^{2}$.

Simplify: ⓐ $\left(3 \sqrt{7}\right) \left(10 \sqrt{7}\right)$ ⓑ $\left(−4 \sqrt{6}\right)^{2}$.

Simplify: ⓐ $3 \left(5 - \sqrt{2}\right)$ ⓑ $\sqrt{2} \left(4 - \sqrt{10}\right)$.

Simplify: ⓐ $2 \left(3 - \sqrt{5}\right)$ ⓑ $\sqrt{3} \left(2 - \sqrt{18}\right)$.

Simplify: ⓐ $6 \left(2 + \sqrt{6}\right)$ ⓑ $\sqrt{7} \left(1 + \sqrt{14}\right)$.

Simplify: ⓐ $\sqrt{5} \left(7 + 2 \sqrt{5}\right)$ ⓑ $\sqrt{6} \left(\sqrt{2} + \sqrt{18}\right)$.

Simplify: ⓐ $\sqrt{6} \left(1 + 3 \sqrt{6}\right)$ ⓑ $\sqrt{12} \left(\sqrt{3} + \sqrt{24}\right)$.

Simplify: ⓐ $\sqrt{8} \left(2 - 5 \sqrt{8}\right)$ ⓑ $\sqrt{14} \left(\sqrt{2} + \sqrt{42}\right)$.

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

Simplify: $\left(1 + \sqrt{6}\right) \left(3 - \sqrt{6}\right)$.

Simplify: $\left(4 - \sqrt{10}\right) \left(2 + \sqrt{10}\right)$.

Simplify: $\left(3 - 2 \sqrt{7}\right) \left(4 - 2 \sqrt{7}\right)$.

Simplify: $\left(6 - 3 \sqrt{7}\right) \left(3 + 4 \sqrt{7}\right)$.

Simplify: $\left(2 - 3 \sqrt{11}\right) \left(4 - \sqrt{11}\right)$.

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: $\left(4 - 2 \sqrt{x}\right) \left(1 + 3 \sqrt{x}\right)$.

Simplify: $\left(6 - 5 \sqrt{m}\right) \left(2 + 3 \sqrt{m}\right)$.

Simplify: $\left(10 + 3 \sqrt{n}\right) \left(1 - 5 \sqrt{n}\right)$.

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: $\left(2 + 5 \sqrt{m}\right)^{2}$.

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

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

Simplify: $\left(1 + \sqrt{5}\right) \left(1 - \sqrt{5}\right)$.

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.

ⓐ $\sqrt{2} \cdot \sqrt{8}$ ⓑ $\left(3 \sqrt{3}\right) \left(2 \sqrt{18}\right)$

Simplify.

ⓐ $\sqrt{7} \cdot \sqrt{14}$ ⓑ $\left(4 \sqrt{8}\right) \left(5 \sqrt{8}\right)$

Simplify.

$\left(5 \sqrt{2}\right) \left(3 \sqrt{6}\right)$

Simplify.

$\left(−2 \sqrt{3}\right) \left(3 \sqrt{18}\right)$

Simplify.

$\left(5 \sqrt{6}\right) \left(− \sqrt{12}\right)$

Simplify.

$\left(−2 \sqrt{7}\right) \left(−2 \sqrt{14}\right)$

Simplify.

ⓐ $\left(\sqrt{15 y}\right) \left(\sqrt{5 y^{3}}\right)$ ⓑ $\left(\sqrt{2 n^{2}}\right) \left(\sqrt{18 n^{3}}\right)$

Simplify.

ⓐ $\left(\sqrt{16 y^{2}}\right) \left(\sqrt{8 y^{4}}\right)$ ⓑ $\left(\sqrt{11 s^{6}}\right) \left(\sqrt{11 s}\right)$

Simplify.

$\left(2 \sqrt{5 b^{3}}\right) \left(4 \sqrt{15 b}\right)$

Simplify.

$\left(5 \sqrt{2 d^{7}}\right) \left(3 \sqrt{50 d^{3}}\right)$

Simplify.

$\left(3 \sqrt{4 y^{4}}\right) \left(3 \sqrt{9 y^{5}}\right)$

Simplify.

$\left(4 \sqrt{2 k^{5}}\right) \left(−3 \sqrt{32 k^{6}}\right)$

Simplify.

ⓐ $\left(\sqrt{11}\right)^{2}$ ⓑ $\left(− \sqrt{21}\right)^{2}$

Simplify.

ⓐ $\left(\sqrt{23}\right)^{2}$
ⓑ $\left(− \sqrt{3}\right)^{2}$

Simplify.

ⓐ $\left(2 \sqrt{13}\right) \left(−9 \sqrt{13}\right)$ ⓑ $\left(6 \sqrt{5}\right)^{2}$

Simplify.

ⓐ $\left(−7 \sqrt{5}\right) \left(−3 \sqrt{10}\right)$ ⓑ $\left(−2 \sqrt{14}\right)^{2}$

Simplify.

ⓐ $4 \left(6 - \sqrt{11}\right)$ ⓑ $\sqrt{2} \left(5 - \sqrt{12}\right)$

Simplify.

ⓐ $7 \left(−2 - \sqrt{11}\right)$ ⓑ $\sqrt{7} \left(6 - \sqrt{14}\right)$

Simplify.

ⓐ $\sqrt{11} \left(8 + 4 \sqrt{11}\right)$ ⓑ $\sqrt{3} \left(\sqrt{12} + \sqrt{27}\right)$

Simplify.

ⓐ $\sqrt{2} \left(−5 + 9 \sqrt{2}\right)$ ⓑ $\sqrt{7} \left(\sqrt{3} - \sqrt{21}\right)$

Simplify.

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

Simplify.

$\left(9 - \sqrt{2}\right) \left(6 + \sqrt{2}\right)$

Simplify.

$\left(5 - \sqrt{7}\right) \left(4 - \sqrt{7}\right)$

Simplify.

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

Simplify.

$\left(\sqrt{11} + \sqrt{5}\right) \left(\sqrt{11} + 6 \sqrt{5}\right)$

Simplify.

$\left(4 \sqrt{6} + 7 \sqrt{13}\right) \left(8 \sqrt{6} - 3 \sqrt{13}\right)$

Simplify.

$\left(9 - \sqrt{w}\right) \left(2 + \sqrt{w}\right)$

Simplify.

$\left(6 + 5 \sqrt{n}\right) \left(11 + 3 \sqrt{n}\right)$

Simplify.

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

Simplify.

ⓐ $\left(5 - \sqrt{10}\right)^{2}$ ⓑ $\left(8 + 3 \sqrt{2}\right)^{2}$

Simplify.

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

Simplify.

$\left(7 + \sqrt{10}\right) \left(7 - \sqrt{10}\right)$

Simplify.

$\left(1 + 8 \sqrt{2}\right) \left(1 - 8 \sqrt{2}\right)$

Simplify.

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

Simplify.

$\left(4 \sqrt{6}\right) \left(− \sqrt{18}\right)$

Simplify.

$\left(−5 \sqrt{7}\right) \left(6 \sqrt{21}\right)$

Simplify.

$\left(\sqrt{35 y^{3}}\right) \left(\sqrt{7 y^{3}}\right)$

Simplify.

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

Simplify.

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

A garden will be made so as to contain two square sections, one section with side length $\sqrt{5} + \sqrt{6}$ yards and one section with side length $\sqrt{2} + \sqrt{3}$ yards. Simplify $\left(\sqrt{5} + \sqrt{6}\right)^{2} + \left(\sqrt{2} + \sqrt{3}\right)^{2}$ to determine the total area of the garden. Round your answer to the nearest tenth.

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