8th Grade - Square Roots and Cube Roots

Introduction

  • In the Laws of Exponents lesson, we learned how to raise a number to a power. Finding a square root of that power is a reversal of that operation.
  • is called a radical sign.
  • Any expression with radical sign values is called a Radical Expression.
  • In 273, 27 is called a radicand, and 3 is called a root number, also sometimes referred to as an index.

Square Roots

  • 32 is 9, the square root of 9 represented as 9 is 3.
  • If a radical number has no root number or index, it can be considered a square root.
  • The square root of an x is nothing but x12

Cube Roots

  • 3 raised to the power of 3, also written as 33 is 27. The reversal of that is the cube root of 27, also written as 273 is 3.

Determining if the Root is Positive or Negative

  • Exponents are nothing but repeated multiplication. Roots are the exact opposite of that.
  • 3 × 3 = 9, and -3  × -3 is also 9. So it is safe to say 9 is ±3 (as we are not sure if 9 results from the multiplication of two positive 3s or two negative 3s.
  • A number multiplied by itself can never have a negative result. If a number is positive, its square is always positive. If a number is negative, its square will also be a positive number. So it is safe to say -9 is not a real number.

The Root of Non-perfect Squares

  • To find the root of the non-perfect square, break the radicand and see if any of the factors are a perfect square.
  • For example, 50 can be written as 25 × 2, which can be further simplified as 52.
  • The square root of a prime number is an irrational number.

Examples

  • 36 is ±6 as 6 × 6, and -6 × -6 both equals to 36.
  • 643 is 4 as 4 multiplied by itself 3 times also written as 43 is 64.
  • 325 is 2 as 25 is 32.

Cheat Sheet

  • A positive number will always have two square roots, one positive and one negative.
  • 1 is ±1
  • The square root of a prime number is an irrational number.
  • 72 = 36 × 2 = 36 . 2 = 6 .2 = 62
  • 43 = 413
  • m2 =± m
  • x3 =  x2 . x = xx

Blunder Area

  • In case you want a root of the non-perfect square, break the radicand and see if any of the factors are a perfect square.