9.4 Multiply square roots

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

ⓐ $\sqrt{6}·\sqrt{6}$
ⓑ $\left(3\sqrt{2}\right)\left(2\sqrt{32}\right)$

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

ⓐ $\sqrt{6}·\sqrt{12}$
ⓑ $\left(2\sqrt{5}\right)\left(2\sqrt{10}\right)$

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

$\left(2\sqrt{3}\right)\left(4\sqrt{6}\right)$

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

$\left(\mathrm{-4}\sqrt{5}\right)\left(5\sqrt{10}\right)$

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

$\left(6\sqrt{2}\right)\left(\text{−}\sqrt{10}\right)$

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

$\left(\mathrm{-2}\sqrt{11}\right)\left(\mathrm{-4}\sqrt{22}\right)$

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

ⓐ $\left(\sqrt{14x^3}\right)\left(\sqrt{7x^3}\right)$
ⓑ $\left(\sqrt{3q^2}\right)\left(\sqrt{48q^3}\right)$

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

ⓐ $\left(\sqrt{8x^3}\right)\left(\sqrt{3x}\right)$
ⓑ $\left(\sqrt{7r}\right)\left(\sqrt{7r^8}\right)$

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

$\left(3\sqrt{8c^5}\right)\left(2\sqrt{6c^3}\right)$

$\left(6\sqrt{3d^3}\right)\left(4\sqrt{12d^5}\right)$

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

$\left(6\sqrt{3d^3}\right)\left(4\sqrt{12d^5}\right)$

$\left(\mathrm{-2}\sqrt{7z^3}\right)\left(3\sqrt{14z^8}\right)$

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

ⓐ ${\left(\sqrt{7}\right)}^2$
ⓑ ${\left(\text{−}\sqrt{15}\right)}^2$

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

ⓐ ${\left(\sqrt{19}\right)}^2$
ⓑ ${\left(\text{−}\sqrt{5}\right)}^2$

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

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

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

ⓐ $\left(\mathrm{-3}\sqrt{12}\right)\left(\mathrm{-2}\sqrt{6}\right)$
ⓑ ${\left(\mathrm{-4}\sqrt{10}\right)}^2$

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

ⓐ $3\left(4-\sqrt{3}\right)$
ⓑ $\sqrt{2}\left(4-\sqrt{6}\right)$

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

ⓐ $5\left(3-\sqrt{7}\right)$
ⓑ $\sqrt{3}\left(4-\sqrt{15}\right)$

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

ⓐ $\sqrt{7}\left(5+2\sqrt{7}\right)$
ⓑ $\sqrt{5}\left(\sqrt{10}+\sqrt{18}\right)$

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

ⓐ $\sqrt{11}\left(\mathrm{-3}+4\sqrt{11}\right)$
ⓑ $\sqrt{3}\left(\sqrt{15}-\sqrt{18}\right)$

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

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

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

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

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

$\left(3-\sqrt{7}\right)\left(5-\sqrt{7}\right)$

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

$\left(1+3\sqrt{10}\right)\left(5-2\sqrt{10}\right)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\sqrt{3}·\sqrt{21}$

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

$\left(\mathrm{-5}+\sqrt{7}\right)\left(6+\sqrt{21}\right)$

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

$\left(\mathrm{-4}\sqrt{2}\right)\left(2\sqrt{18}\right)$

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

$\left(4\sqrt{12x^5}\right)\left(2\sqrt{6x^3}\right)$

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

$\left(\mathrm{-4}\sqrt{17}\right)\left(\mathrm{-3}\sqrt{17}\right)$

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

A landscaper wants to put a square reflecting pool next to a triangular deck, as shown below. The triangular deck is a right triangle, with legs of length 9 feet and 11 feet, and the pool will be adjacent to the hypotenuse.

1. ⓐ Use the Pythagorean Theorem to find the length of a side of the pool. Round your answer to the nearest tenth of a foot.
2. ⓑ Find the exact area of the pool.

An artist wants to make a small monument in the shape of a square base topped by a right triangle, as shown below. The square base will be adjacent to one leg of the triangle. The other leg of the triangle will measure 2 feet and the hypotenuse will be 5 feet.

1. ⓐ Use the Pythagorean Theorem to find the length of a side of the square base. Round your answer to the nearest tenth of a foot.
   
2. ⓑ Find the exact area of the face of the square base.

A square garden will be made with a stone border on one edge. If only $3+\sqrt{10}$ feet of stone are available, simplify ${\left(3+\sqrt{10}\right)}^2$ to determine the area of the largest such garden.

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 $\left(\sqrt{5}+\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)$ to determine the total area of the garden.

Suppose a third section will be added to the garden in the previous exercise. The third section is to have a width of $\sqrt{432}$ feet. Write an expression that gives the total area of the garden.

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

Use the binomial square pattern to simplify ${\left(3+\sqrt{2}\right)}^2$ . Explain all your steps.