Multi-Step Radical Expressions Solutions

• Apply the same rules for exponents and radicals when working with    multi-step expressions    containing radicals.
• Use    order of operations    to combine terms in the correct order. 
• Remember: $a\sqrt{b}·c\sqrt{d}=ac\sqrt{bd}$
  
•    Coefficients    or degree zero terms (constants) can be simplified.
•    Radicands    can be simplified.
• Terms    outside   the radical cannot be combined with terms    inside    the radical.
• To combine terms with a radical, the    same index    and the    same radicand    are necessary. 

Example 7

Simplify.

$\sqrt{3}\left(7\sqrt{6}–5\right)–6\sqrt{2}$

Example 8

Find the area of a rectangle with sides $\mathbf{3}\sqrt{\mathbf{5}}\mathbf{–}\mathbf{2}\sqrt{\mathbf{10}}$ units and $\mathbf{3}\sqrt{\mathbf{5}}\mathbf{–}\sqrt{\mathbf{2}}$ units.

A=lw

Example 9

Find the base of a triangle with an area of $\mathbf{2}\sqrt{\mathbf{2}}\mathbf{+}\mathbf{2}\sqrt{\mathbf{5}}$ square units and a height of$\sqrt{\mathbf{5}}\mathbf{+}\sqrt{\mathbf{2}}$units. Write answers in simplified radical form.