Roots and Radicals

Multiply Complex Numbers

$\sqrt{-4}.\sqrt{-16}$

Multiply Complex Numbers

$(5 -\sqrt{-12}) (-3 +\sqrt{-75})$

In the following exercises, multiply using the Product of Binomial Squares Pattern.

In the following exercises, multiply using the Product of Complex Conjugates Pattern. 

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

Divide Complex Numbers

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

In the following exercises, simplify. 

$i^{48}$

In the following exercises, simplify.

$i^{255}$

Simplify using absolute values as necessary.

$\sqrt{169x^8{y}^6}$

Simplify using absolute values as necessary.

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

Simplify using absolute values as necessary.

$\sqrt{\frac{45x^3{y}^4}{180x^5{y}^2}}$

In the following exercises, simplify. Assume all variables are positive.

(a) ${216}^{-\frac{1}{4}}$

(b) $-{49}^{\frac{3}{2}}$

In the following exercises, simplify. Assume all variables are positive.

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

In the following exercises, simplify. Assume all variables are positive.

${\left(\frac{8x^{{\displaystyle \frac{2}{3}}}{y}^{-{\displaystyle \frac{5}{2}}}}{x^{-\frac{7}{3}}{y}^{\frac{1}{2}}}\right)}^{\frac{1}{3}}$

Simplify. Assume all variables are positive.

$\sqrt{27x^2} -4x\sqrt{12} + \sqrt{108x^2}$

In the following exercises, simplify. Assume all variables are positive.

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

In the following exercises, simplify. Assume all variables are positive.

$\sqrt[3]{4}\left(\sqrt[3]{16}-\sqrt[3]{6}\right)$

In the following exercises, simplify. Assume all variables are positive.

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

In the following exercises, simplify. Assume all variables are positive.

$\frac{\sqrt[3]{128}}{\sqrt[3]{54}}$

In the following exercises, simplify. Assume all variables are positive.

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

In the following exercises, simplify. Assume all variables are positive.

$\frac{1}{\sqrt[3]{5}}.$

In the following exercises, simplify. Assume all variables are positive.

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

In the following exercises, simplify. Assume all variables are positive.

$\sqrt{-4}.\sqrt{-9}$

In the following exercises, simplify. Assume all variables are positive.

$-4i\left(-2-3i\right)$

In the following exercises, simplify. Assume all variables are positive.

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

In the following exercises, simplify. Assume all variables are positive.

i¹⁷²$i^{172}.$

In the following exercises, solve.

$\sqrt{2x+5}+8=6$

In the following exercises, solve.

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

In the following exercises, solve.

$\sqrt[3]{2x^2-6x-23}=\sqrt[3]{x^2-3x+5}$

In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g\left(x\right)=\sqrt{x+2}$

In the following exercises, simplify. 

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