Honors Geometry: Introduction to Simplifying Radicals

Fundamental Concepts and Rules of Radicals

  • Perfect Squares: A perfect square is an integer generated by multiplying a whole number by itself (n2n^2).

  • Radicand: The number or algebraic expression located inside the radical symbol (radicand\sqrt{\text{radicand}}).

  • Rule of Principal Square Roots: When finding the solution to a square root problem involving a perfect square, the evaluated result must be written as a rational integer. The final answer is NEVER written inside a radical sign.

  • Factoring Order Standard: When factoring a radicand into a perfect square factor and another factor, ALWAYS write the perfect square number first (e.g., 18=9⋅218 = 9 \cdot 2).

  • Strategy for Radical Simplification:

    • Factor out the largest perfect square factor contained within the radicand.

    • Take the square root of the perfect square factor and write its evaluated integer value outside the radical.

    • Leave the non-perfect square factor inside the radical.

    • If a smaller perfect square factor is selected initially, the remaining radicand must be factored again until no perfect square factors remain inside the radical.

  • Handling Existing Coefficients: If a radical term already contains a coefficient outside the radical symbol (e.g., cac\sqrt{a}), simplify the inner radical first, then multiply the resulting root by the existing outside coefficient (c⋅perfect square⋅remainderc \cdot \sqrt{\text{perfect square}} \cdot \sqrt{\text{remainder}}).

  • Rules for Adding and Subtracting Radicals:

    • Radicals can only be added or subtracted if their radicands are identical ("like radicals").

    • If radicands are not identical, simplify each radical term fully first.

    • If radicands remain different after full simplification, the terms cannot be combined into a single radical term and must remain separate.

  • Rules for Radical Fractions and Rationalizing Denominators:

    • If a radical contains a fraction underneath it, apply the quotient property of radicals: ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}.

    • Expressions are not considered fully simplified if a radical remains in the denominator.

    • Rationalizing the Denominator: Eliminate radicals in the denominator by multiplying both the numerator and denominator by the square root present in the denominator.

First Twelve Perfect Squares

  • 12=11^2 = 1

  • 22=42^2 = 4

  • 32=93^2 = 9

  • 42=164^2 = 16

  • 52=255^2 = 25

  • 62=366^2 = 36

  • 72=497^2 = 49

  • 82=648^2 = 64

  • 92=819^2 = 81

  • 102=10010^2 = 100

  • 112=12111^2 = 121

  • 122=14412^2 = 144

  • Complete Sequence of First Twelve Perfect Squares: 1,4,9,16,25,36,49,64,81,100,121,1441, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Evaluating Basic Square Roots

  • Problem 1: 121=11\sqrt{121} = 11

  • Problem 2: 25=5\sqrt{25} = 5

  • Problem 3: 9=3\sqrt{9} = 3

  • Problem 4: 16=4\sqrt{16} = 4

  • Problem 5: 4=2\sqrt{4} = 2

  • Problem 6: 144=12\sqrt{144} = 12

  • Problem 7: 1=1\sqrt{1} = 1

  • Problem 8: 36=6\sqrt{36} = 6

  • Problem 9: 81=9\sqrt{81} = 9

  • Problem 10: 100=10\sqrt{100} = 10

  • Problem 11: 49=7\sqrt{49} = 7

  • Problem 12: 64=8\sqrt{64} = 8

Factoring Numbers with Perfect Squares

  • Problem 13: 18=9⋅218 = 9 \cdot 2

  • Problem 14: 24=4⋅624 = 4 \cdot 6

  • Problem 15: 75=25⋅375 = 25 \cdot 3

  • Problem 16: 12=4⋅312 = 4 \cdot 3

  • Problem 17: 300=100⋅3300 = 100 \cdot 3

  • Problem 18: 8=4⋅28 = 4 \cdot 2

  • Problem 19: 32=16⋅232 = 16 \cdot 2

  • Problem 20: 50=25⋅250 = 25 \cdot 2

  • Problem 21: 94=1⋅9494 = 1 \cdot 94 (contains no perfect square factor greater than 11)

  • Problem 22: 27=9⋅327 = 9 \cdot 3

  • Problem 23: 28=4⋅728 = 4 \cdot 7

  • Problem 24: 45=9⋅545 = 9 \cdot 5

Simplifying Radical Expressions Step-by-Step

  • Problem 27: 12=4⋅3=23\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}

  • Problem 28: 45=9⋅5=35\sqrt{45} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}

  • Problem 29: 28=4⋅7=27\sqrt{28} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}

  • Problem 30: 75=25⋅3=53\sqrt{75} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}

  • Problem 31: 200=100⋅2=102\sqrt{200} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}

  • Problem 32: 50=25⋅2=52\sqrt{50} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}

  • Problem 33: 212=24⋅3=2(2)3=432\sqrt{12} = 2\sqrt{4} \cdot \sqrt{3} = 2(2)\sqrt{3} = 4\sqrt{3}

  • Problem 34: 418=49⋅2=4(3)2=1224\sqrt{18} = 4\sqrt{9} \cdot \sqrt{2} = 4(3)\sqrt{2} = 12\sqrt{2}

Adding and Subtracting Radicals

  • Problem 35: 35+25=(3+2)5=553\sqrt{5} + 2\sqrt{5} = (3 + 2)\sqrt{5} = 5\sqrt{5}

  • Problem 36: 7+5=7+5\sqrt{7} + \sqrt{5} = \sqrt{7} + \sqrt{5} (Radicands 77 and 55 are different and fully simplified; terms cannot be combined)

  • Problem 37: 10+10=110+110=210\sqrt{10} + \sqrt{10} = 1\sqrt{10} + 1\sqrt{10} = 2\sqrt{10}

  • Problem 38: 20+20=4⋅5+4⋅5=25+25=45\sqrt{20} + \sqrt{20} = \sqrt{4 \cdot 5} + \sqrt{4 \cdot 5} = 2\sqrt{5} + 2\sqrt{5} = 4\sqrt{5}

  • Problem 39: 227+12=29⋅3+4⋅3=2(3)3+23=63+23=832\sqrt{27} + \sqrt{12} = 2\sqrt{9 \cdot 3} + \sqrt{4 \cdot 3} = 2(3)\sqrt{3} + 2\sqrt{3} = 6\sqrt{3} + 2\sqrt{3} = 8\sqrt{3}

  • Problem 40: 58+32=54⋅2+16⋅2=5(2)2+42=102+42=1425\sqrt{8} + \sqrt{32} = 5\sqrt{4 \cdot 2} + \sqrt{16 \cdot 2} = 5(2)\sqrt{2} + 4\sqrt{2} = 10\sqrt{2} + 4\sqrt{2} = 14\sqrt{2}

Radical Fractions and Rationalizing Denominators

  • Problem 41: 425=425=25\sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5}

  • Problem 42: 37=37=3⋅77⋅7=217\sqrt{\frac{3}{7}} = \frac{\sqrt{3}}{\sqrt{7}} = \frac{\sqrt{3} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} = \frac{\sqrt{21}}{7}

  • Problem 43: 13=1⋅33⋅3=33\frac{1}{\sqrt{3}} = \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}

  • Problem 44: 105=10⋅55⋅5=1055=25\frac{10}{\sqrt{5}} = \frac{10 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}

Comprehensive Practice Solutions

  • Problem 45: 63=9⋅7=37\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}

  • Problem 46: 80=16⋅5=45\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}

  • Problem 47: 500=100⋅5=105\sqrt{500} = \sqrt{100 \cdot 5} = 10\sqrt{5}

  • Problem 48: 98=49⋅2=72\sqrt{98} = \sqrt{49 \cdot 2} = 7\sqrt{2}

  • Problem 49: 54=9⋅6=36\sqrt{54} = \sqrt{9 \cdot 6} = 3\sqrt{6}

  • Problem 50: 72=36⋅2=62\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}

  • Problem 51: 40=4⋅10=210\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}

  • Problem 52: 60=4⋅15=215\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}

  • Problem 53: 30=30\sqrt{30} = \sqrt{30} (No perfect square factors greater than 11; expression is fully simplified)

  • Problem 54: 724=74⋅6=7⋅26=1467\sqrt{24} = 7\sqrt{4 \cdot 6} = 7 \cdot 2\sqrt{6} = 14\sqrt{6}

  • Problem 55: 318+48=39⋅2+44⋅2=3(3)2+4(2)2=92+82=1723\sqrt{18} + 4\sqrt{8} = 3\sqrt{9 \cdot 2} + 4\sqrt{4 \cdot 2} = 3(3)\sqrt{2} + 4(2)\sqrt{2} = 9\sqrt{2} + 8\sqrt{2} = 17\sqrt{2}

  • Problem 56: 20+18=4⋅5+9⋅2=25+32\sqrt{20} + \sqrt{18} = \sqrt{4 \cdot 5} + \sqrt{9 \cdot 2} = 2\sqrt{5} + 3\sqrt{2} (Radicands 55 and 22 are different; expression cannot be combined further)

  • Problem 57: 212+518=24⋅3+59⋅2=2(2)3+5(3)2=43+1522\sqrt{12} + 5\sqrt{18} = 2\sqrt{4 \cdot 3} + 5\sqrt{9 \cdot 2} = 2(2)\sqrt{3} + 5(3)\sqrt{2} = 4\sqrt{3} + 15\sqrt{2} (Radicands 33 and 22 are different; expression cannot be combined further)

  • Problem 58: 36=36=12=12=1⋅22⋅2=22\frac{\sqrt{3}}{\sqrt{6}} = \sqrt{\frac{3}{6}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}

  • Problem 59: 152=1⋅252⋅2=25⋅2=210\frac{1}{5\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{5\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{5 \cdot 2} = \frac{\sqrt{2}}{10}

  • Problem 60: 232=23⋅22⋅2=262=6\frac{2\sqrt{3}}{\sqrt{2}} = \frac{2\sqrt{3} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{6}}{2} = \sqrt{6}