12 multiple-choice questions, progressively harder.
Rationalize and simplify 65+1\frac{6}{\sqrt{5} + 1}5+16.
Solution
Correct answer: C
Multiply by the conjugate 5−1\sqrt{5} - 15−1. The denominator is 5−1=45 - 1 = 45−1=4 and the numerator is 6(5−1)6(\sqrt{5} - 1)6(5−1).
65+1=6(5−1)5−1=6(5−1)4=3(5−1)2\frac{6}{\sqrt{5} + 1} = \frac{6(\sqrt{5} - 1)}{5 - 1} = \frac{6(\sqrt{5} - 1)}{4} = \frac{3(\sqrt{5} - 1)}{2}5+16=5−16(5−1)=46(5−1)=23(5−1)
The 666 over 444 reduces to 32\frac{3}{2}23.
Rationalize 17+3\frac{1}{\sqrt{7} + \sqrt{3}}7+31.
Correct answer: A
Multiply by the conjugate 7−3\sqrt{7} - \sqrt{3}7−3. The denominator is 7−3=47 - 3 = 47−3=4.
17+3=7−37−3=7−34\frac{1}{\sqrt{7} + \sqrt{3}} = \frac{\sqrt{7} - \sqrt{3}}{7 - 3} = \frac{\sqrt{7} - \sqrt{3}}{4}7+31=7−37−3=47−3
The difference of squares gives 444, not 101010.
Rationalize 14−7\frac{1}{4 - \sqrt{7}}4−71.
Correct answer: D
Multiply by the conjugate 4+74 + \sqrt{7}4+7. The denominator is 42−(7)2=16−7=94^2 - (\sqrt{7})^2 = 16 - 7 = 942−(7)2=16−7=9.
14−7=4+716−7=4+79\frac{1}{4 - \sqrt{7}} = \frac{4 + \sqrt{7}}{16 - 7} = \frac{4 + \sqrt{7}}{9}4−71=16−74+7=94+7
The numerator takes the opposite sign, 4+74 + \sqrt{7}4+7.
Rationalize 15−3\frac{1}{\sqrt{5} - 3}5−31.
Correct answer: B
Multiply by the conjugate 5+3\sqrt{5} + 35+3. The denominator is (5)2−32=5−9=−4(\sqrt{5})^2 - 3^2 = 5 - 9 = -4(5)2−32=5−9=−4.
15−3=5+35−9=5+3−4=−5+34\frac{1}{\sqrt{5} - 3} = \frac{\sqrt{5} + 3}{5 - 9} = \frac{\sqrt{5} + 3}{-4} = -\frac{\sqrt{5} + 3}{4}5−31=5−95+3=−45+3=−45+3
The negative denominator makes the result negative.
Rationalize 110+6\frac{1}{\sqrt{10} + \sqrt{6}}10+61.
Multiply by the conjugate 10−6\sqrt{10} - \sqrt{6}10−6. The denominator is 10−6=410 - 6 = 410−6=4.
110+6=10−610−6=10−64\frac{1}{\sqrt{10} + \sqrt{6}} = \frac{\sqrt{10} - \sqrt{6}}{10 - 6} = \frac{\sqrt{10} - \sqrt{6}}{4}10+61=10−610−6=410−6
The difference of squares gives 444, not 161616.
Rationalize and simplify 550\frac{5}{\sqrt{50}}505.
Simplify the radical first, since 50=52\sqrt{50} = 5\sqrt{2}50=52.
550=552=12=22\frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}505=525=21=22
The 555 over 555 cancels, and 12\frac{1}{\sqrt{2}}21 rationalizes to 22\frac{\sqrt{2}}{2}22.
Rationalize 132−4\frac{1}{3\sqrt{2} - 4}32−41.
Multiply by the conjugate 32+43\sqrt{2} + 432+4. The denominator is (32)2−42=18−16=2(3\sqrt{2})^2 - 4^2 = 18 - 16 = 2(32)2−42=18−16=2.
132−4=32+418−16=32+42\frac{1}{3\sqrt{2} - 4} = \frac{3\sqrt{2} + 4}{18 - 16} = \frac{3\sqrt{2} + 4}{2}32−41=18−1632+4=232+4
Note (32)2=9⋅2=18(3\sqrt{2})^2 = 9\cdot 2 = 18(32)2=9⋅2=18.
Rationalize 12−5\frac{1}{2 - \sqrt{5}}2−51.
Multiply by the conjugate 2+52 + \sqrt{5}2+5. The denominator is 22−(5)2=4−5=−12^2 - (\sqrt{5})^2 = 4 - 5 = -122−(5)2=4−5=−1.
12−5=2+54−5=2+5−1=−2−5\frac{1}{2 - \sqrt{5}} = \frac{2 + \sqrt{5}}{4 - 5} = \frac{2 + \sqrt{5}}{-1} = -2 - \sqrt{5}2−51=4−52+5=−12+5=−2−5
Dividing by −1-1−1 flips the sign of each term.
Rationalize and simplify 26−2\frac{\sqrt{2}}{\sqrt{6} - \sqrt{2}}6−22.
Multiply by the conjugate 6+2\sqrt{6} + \sqrt{2}6+2. The denominator is 6−2=46 - 2 = 46−2=4, and the numerator is 2(6+2)=12+2=23+2\sqrt{2}(\sqrt{6} + \sqrt{2}) = \sqrt{12} + 2 = 2\sqrt{3} + 22(6+2)=12+2=23+2.
26−2=23+26−2=23+24=3+12\frac{\sqrt{2}}{\sqrt{6} - \sqrt{2}} = \frac{2\sqrt{3} + 2}{6 - 2} = \frac{2\sqrt{3} + 2}{4} = \frac{\sqrt{3} + 1}{2}6−22=6−223+2=423+2=23+1
Simplify 12=23\sqrt{12} = 2\sqrt{3}12=23, then divide top and bottom by 222.
Which of these expressions is NOT fully rationalized (still has a square root in the denominator)?
An expression is fully rationalized only when no square root remains in the denominator.
23=233\frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}32=323
Only 23\frac{2}{\sqrt{3}}32 still has a root on the bottom, so it is not yet rationalized.
Rationalize and simplify 11+5\frac{1}{1 + \sqrt{5}}1+51.
Multiply by the conjugate 1−51 - \sqrt{5}1−5. The denominator is 12−(5)2=1−5=−41^2 - (\sqrt{5})^2 = 1 - 5 = -412−(5)2=1−5=−4.
11+5=1−51−5=1−5−4=5−14\frac{1}{1 + \sqrt{5}} = \frac{1 - \sqrt{5}}{1 - 5} = \frac{1 - \sqrt{5}}{-4} = \frac{\sqrt{5} - 1}{4}1+51=1−51−5=−41−5=45−1
Multiplying numerator and denominator by −1-1−1 turns 1−5−4\frac{1 - \sqrt{5}}{-4}−41−5 into 5−14\frac{\sqrt{5} - 1}{4}45−1.
Rationalize and simplify 107+2\frac{10}{\sqrt{7} + \sqrt{2}}7+210.
Multiply by the conjugate 7−2\sqrt{7} - \sqrt{2}7−2. The denominator is 7−2=57 - 2 = 57−2=5 and the numerator is 10(7−2)10(\sqrt{7} - \sqrt{2})10(7−2).
107+2=10(7−2)7−2=10(7−2)5=27−22\frac{10}{\sqrt{7} + \sqrt{2}} = \frac{10(\sqrt{7} - \sqrt{2})}{7 - 2} = \frac{10(\sqrt{7} - \sqrt{2})}{5} = 2\sqrt{7} - 2\sqrt{2}7+210=7−210(7−2)=510(7−2)=27−22
The 101010 over 555 reduces to 222.
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