Questions on simplifying expressions with square root radicals are presented. The answers to the questions are at the bottom of the page and the solutions with full explanations to these questions are also included.
\( \sqrt{\dfrac{x} {y}} = \dfrac{\sqrt x}{\sqrt y} \) for \( x \ge 0 \) and \( y \ge 0 \)
Example: \( \sqrt{\dfrac{1} {36}} = \dfrac{\sqrt 1}{\sqrt 36} = \dfrac{1}{6} \)
We can group expressions with the same radicand (the term under the radical)
Example: \( 5 \sqrt 3 + 8 \sqrt 3 = (5 + 8) \sqrt 3 = 13 \sqrt 3 \)
Example: The expression \( 2 \sqrt 5 + 8 \sqrt 3 \) cannot be simplified because the radicands \( 5 \) and \( 3 \) are different.
Common Mistakes to Avoid when Working with Radicals
1) \( \sqrt{x + y} \) IS NOT EQUAL TO \( \sqrt x + \sqrt y \)
Example: \( \sqrt{16 + 9} = \sqrt{25} = 5 \) IS NOT EQUAL TO \( \sqrt {16} + \sqrt 9 = 4 + 3 = 7 \)
2) \( \sqrt{x - y} \) IS NOT EQUAL TO \( \sqrt x - \sqrt y \)
Example: \( \sqrt{25 - 16} = \sqrt{9} = 3 \) IS NOT EQUAL TO \( \sqrt {25} - \sqrt {16} = 5 - 4 = 1 \)
Questions to Simplify Square Root Radicals
Simplify the expression \( 2 \sqrt{50} + 12 \sqrt{8} \).
A) \( 14 \sqrt{58} \)
B) 240
C) 280
D) 98
E) \( 34 \sqrt 2 \)
Simplify the expression \( \sqrt{27} - \sqrt{300} \).
A) \( \sqrt{-273} \)
B) \( -7 \sqrt 3 \)
C) -147
D) \( 3 - 10 \sqrt3 \)
E) \( 7\sqrt3 \)
Simplify the expression \( - 2 \sqrt{16y} + 10 \sqrt y \).
A) \( 8 \sqrt y \)
B) \( 8 \sqrt{-15y} \)
C) \( 8 \sqrt{17y} \)
D) \( 2 \sqrt{16y} \)
E) \( 2 \sqrt y \)