Lecture Notes: Square Roots and Cube Roots

Square Roots

• Definition: A square root of a number is a value that, when multiplied by itself, gives the original number.

  • Example: Square root of 25 is 5 because 5 x 5 = 25.

• Negative Square Roots:

  • Example: Negative square root of 25 is -5.

• Square Roots of Negative Numbers:

  • Square root of negative numbers involves imaginary numbers.
  • Example: Square root of -25 is 5i, where i represents the square root of -1.

• Practice Problems:

  • Square root of 81: 9
  • Negative square root of 81: -9
  • Square root of -81: 9i

• Square Roots of Fractions:

  • Example: Square root of ( \frac{36}{49} ) is ( \frac{6}{7} ).
  • Negative square root of ( \frac{121}{169} ): -( \frac{11}{13} ).
  • Negative square root of ( \frac{-81}{144} ): ( \frac{3}{4}i ).

• Square Roots of Decimals:

  • Square root of 0.16 is 0.4 because 16 has 2 digits, and the square root should have 1 digit.
  • Square root of 0.0025 is 0.05.
  • Square root of 0.000081 is 0.009.

Cube Roots

• Definition: A cube root of a number is a value that, when used three times in a multiplication, gives the original number.

  • Example: Cube root of 8 is 2 because 2 x 2 x 2 = 8.

• Negative Cube Roots:

  • Example: Cube root of -27 is -3.
  • Cube root of -64 (with a negative outside): -4.

• Cube Roots of Fractions:

  • Example: Cube root of ( \frac{125}{216} ) is ( \frac{5}{6} ).

• Cube Roots of Decimals:

  • Example: Cube root of 0.008 is 0.2.
  • Cube root of 0.000027 is 0.03.
  • Cube root of 0.000000064 is 0.004.

Summary

• Square roots are calculated for positive real numbers, and negative numbers involve imaginary numbers.
• Cube roots can be calculated for both positive and negative numbers without involving imaginary numbers.
• When calculating roots of decimals, the number of digits to the right of the decimal is crucial to determine the digits in the answer.