Module 2: Square Roots, Cube Roots, and Radical Expressions

Core Concepts and Definitions of Roots


Handwritten study notes on square roots, cube roots, and radical evaluation steps.

  • Radical Sign (\sqrt{\quad}): The symbol used to indicate a positive square root.

  • Square Root: A number that, when multiplied by itself (multiplied 2 times), gives the original number.

    • Example: 32=93^2 = 9, so a square root of 99 is 33.

  • Cube Root (3\sqrt[3]{\quad}): A number that, when multiplied by itself 3 times, yields the given quantity.

    • Example: 273=3\sqrt[3]{27} = 3 because 3×3×3=273 \times 3 \times 3 = 27.

  • Principal Square Root: The non-negative (positive) output of a square root.

    • Rule: Unless specifically stated otherwise, always evaluate for the principal square root.

Sign Conventions and Fractional Radicals

  • Sign Conventions:

    • Positive square root: +25=+5+\sqrt{25} = +5 or 25=5\sqrt{25} = 5

    • Negative square root: −25=−5-\sqrt{25} = -5

  • Square Roots of Fractions:

    • 36100=610\sqrt{\frac{36}{100}} = \frac{6}{10}

    • The fraction 36100\frac{36}{100} breaks into evaluating the numerator 36=6\sqrt{36} = 6 and the denominator 100=10\sqrt{100} = 10.

Practice Problems

Square Root Evaluations

  • 81=9\sqrt{81} = 9

  • 225=15\sqrt{225} = 15

  • 121=11\sqrt{121} = 11

  • 1.44=1.2\sqrt{1.44} = 1.2

  • 0.81=0.9\sqrt{0.81} = 0.9

  • 10,000=100\sqrt{10,000} = 100

Cube Root Evaluations

  • 83=2\sqrt[3]{8} = 2

  • −1253=−5\sqrt[3]{-125} = -5

  • 3433=7\sqrt[3]{343} = 7

  • 1,0003=10\sqrt[3]{1,000} = 10

Order of Operations with Radicals

  • Evaluated Expression: 3.5+62−113.5 + \sqrt{6^2 - 11}

    • Step 1 (Exponents inside radical): Compute 62=366^2 = 36.

      • Expression: 3.5+36−113.5 + \sqrt{36 - 11}

    • Step 2 (Subtraction inside radical): Subtract 36−11=2536 - 11 = 25.

      • Expression: 3.5+253.5 + \sqrt{25}

    • Step 3 (Simplify Radical): Evaluate 25=5\sqrt{25} = 5

      • Expression: 3.5+53.5 + 5

    • Step 4 (Final Addition): Add 3.5+53.5 + 5

      • Result: 8.58.5

Roots of Negative Numbers

  • Square Roots of Negative Numbers:

    • −16\sqrt{-16} is not a real number because no real number multiplied by itself yields a negative product.

  • Cube Roots of Negative Numbers:

    • Cube roots of negative numbers are real numbers because multiplying three negative numbers results in a negative product.

    • Demonstration: −33=(−3)×(−3)×(−3)=9×(−3)=−27-3^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27

    • Evaluating the radical: −273=−3\sqrt[3]{-27} = -3

    • Scalar multiplication example: 3−273=3×(−3)=−93\sqrt[3]{-27} = 3 \times (-3) = -9


Sign Conventions and Fractional Radicals

  • Sign Conventions:

    • Positive square root: +25=+5+\sqrt{25} = +5 or 25=5\sqrt{25} = 5

    • Negative square root: −25=−5-\sqrt{25} = -5

  • Square Roots of Fractions:

    • 36100=610\sqrt{\frac{36}{100}} = \frac{6}{10}

    • The fraction 36100\frac{36}{100} breaks into evaluating the numerator 36=6\sqrt{36} = 6 and the denominator 100=10\sqrt{100} = 10.

    • 4964=78\sqrt{\frac{49}{64}} = \frac{7}{8}

    • The fraction 4964\frac{49}{64} breaks into evaluating the numerator 49=7\sqrt{49} = 7 and the denominator 64=8\sqrt{64} = 8.

Practice Problems

Square Root Evaluations

  • 81=9\sqrt{81} = 9

  • 225=15\sqrt{225} = 15

  • 121=11\sqrt{121} = 11

  • 1.44=1.2\sqrt{1.44} = 1.2

  • 0.81=0.9\sqrt{0.81} = 0.9

  • 10,000=100\sqrt{10,000} = 100

  • 49=7\sqrt{49} = 7

  • 169=13\sqrt{169} = 13

  • 0.04=0.2\sqrt{0.04} = 0.2

  • 400=20\sqrt{400} = 20

Cube Root Evaluations

  • 83=2\sqrt[3]{8} = 2

  • −1253=−5\sqrt[3]{-125} = -5

  • 3433=7\sqrt[3]{343} = 7

  • 1,0003=10\sqrt[3]{1,000} = 10

  • 273=3\sqrt[3]{27} = 3

  • −643=−4\sqrt[3]{-64} = -4

  • 2163=6\sqrt[3]{216} = 6

Order of Operations with Radicals

  • Evaluated Expression 1: 3.5+62−113.5 + \sqrt{6^2 - 11}

    • Step 1 (Exponents inside radical): Compute 62=366^2 = 36.

    • Expression: 3.5+36−113.5 + \sqrt{36 - 11}

    • Step 2 (Subtraction inside radical): Subtract 36−11=2536 - 11 = 25.

    • Expression: 3.5+253.5 + \sqrt{25}

    • Step 3 (Simplify Radical): Evaluate 25=5\sqrt{25} = 5

    • Expression: 3.5+53.5 + 5

    • Step 4 (Final Addition): Add 3.5+53.5 + 5

    • Result: 8.58.5

  • Evaluated Expression 2: 10−52−910 - \sqrt{5^2 - 9}

    • Step 1 (Exponents inside radical): Compute 52=255^2 = 25.

    • Expression: 10−25−910 - \sqrt{25 - 9}

    • Step 2 (Subtraction inside radical): Subtract 25−9=1625 - 9 = 16.

    • Expression: 10−1610 - \sqrt{16}

    • Step 3 (Simplify Radical): Evaluate 16=4\sqrt{16} = 4

    • Expression: 10−410 - 4

    • Step 4 (Final Subtraction): Subtract 10−410 - 4

    • Result: 66

Roots of Negative Numbers

  • Square Roots of Negative Numbers:

    • −16\sqrt{-16} is not a real number because no real number multiplied by itself yields a negative product.

    • −25\sqrt{-25} is not a real number.

  • Cube Roots of Negative Numbers:

    • Cube roots of negative numbers are real numbers because multiplying three negative numbers results in a negative product.

    • Demonstration: −33=(−3)×(−3)×(−3)=9×(−3)=−27-3^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27

    • Evaluating the radical: −273=−3\sqrt[3]{-27} = -3

    • Scalar multiplication example: 3−273=3×(−3)=−93\sqrt[3]{-27} = 3 \times (-3) = -9

    • Additional example: −643=−4\sqrt[3]{-64} = -4