Simplify $4x^2 + 5x - 6x^2 + 3x - 2x$.

First, we sort out like variables and reorder them to be combined.

$\begin{array}{ll} & {4x}^{2} + 5x - 6x^{2} + 3x - 2x \\ \text{becomes}& {4x}^{2}-{6x}^{2}\text{ and }5x+3x-2x \end{array}$

Combining like variables yields:

$-2x^2 + 6x$

Simplify $5\sqrt{11} + 5\sqrt{13} - 2\sqrt{13} + 6\sqrt{11} - 2\sqrt{11}$.

First, we sort out like variables and reorder them to be combined.

$\begin{array}{ll} & 5\sqrt{11} + 5\sqrt{13} - 2\sqrt{13} + 6\sqrt{11} - 2\sqrt{11} \\ \text{becomes}& 5\sqrt{13}-2\sqrt{13}\text{ and }5\sqrt{11}+6\sqrt{11}-2\sqrt{11} \end{array}$

Combining like radicals yields:

$3\sqrt{13} + 9\sqrt{11}$

Simplify $4\sqrt{45} + 3\sqrt{18} - \sqrt{98} + 2\sqrt{20}$.

All of these radicals need to be simplified before they can be combined.

$\begin{array}{ll} & 4\sqrt{45}+3\sqrt{18}-\sqrt{98}+2\sqrt{20} \\ \text{becomes} & 4\sqrt{9\cdot 5}+3\sqrt{9\cdot 2} - \sqrt{49\cdot 2}+2\sqrt{5\cdot 4} \\ \text{simplifying to}& 4\cdot3\sqrt{5}+3\cdot 3\sqrt{2}-7\sqrt{2}+2\cdot 2\sqrt{5} \\ \text{and reduces to}&12\sqrt{5}+9\sqrt{2}-7\sqrt{2}+4\sqrt{5} \end{array}$

Recombining these so they can be added and subtracted yields:

$12\sqrt{5}+4\sqrt{5}\text{ and }9\sqrt{2}-7\sqrt{2}$

Combining like radicals yields:

$16\sqrt{5} + 2\sqrt{2}$

Simplify $4\sqrt[3]{54} - 9\sqrt[3]{16} + 5\sqrt[3]{9}$.

Like example 9.3.3, these radicals need to be simplified before they can be combined.

$\begin{array}{ll} & 4 \sqrt[3]{54} - 9 \sqrt[3]{16} + 5 \sqrt[3]{9} \\ \text{becomes} & 4 \sqrt[3]{27\cdot 2} - 9 \sqrt[3]{8\cdot 2} + 5 \sqrt[3]{9} \\ \text{simplifying to} & 4 \cdot 3 \sqrt[3]{2} - 9 \cdot 2 \sqrt[3]{2} + 5 \sqrt[3]{9} \\ \text{and reduces to}& 12 \sqrt[3]{2} - 18 \sqrt[3]{2} + 5 \sqrt[3]{9} \end{array}$

Combining like radicals yields:

$5\sqrt[3]{9} - 6\sqrt[3]{2}$