Square and cube roots Knowledge-Based Study

Study notes

1. What is a Square Root?

   The square root of a number is a value that, when multiplied by itself, yields the original number. For any non-negative number a, the principal (non-negative) square root is denoted by the radical symbol:

   $\sqrt{a} = b$ because $b^2 = a$ and $b \ge 0$

   Perfect Squares

   An integer whose square root is also an integer is called a perfect square.

   • $\sqrt{0} = 0$ because $0^2 = 0$
   • $\sqrt{1} = 1$ because $1^2 = 1$
   • $\sqrt{25} = 5$ because $5^2 = 25$
   • $\sqrt{144} = 12$ because $12^2 = 144$

   Important Note: While both 5 and -5 square to 25, the radical symbol $\sqrt{25}$ designates only the principal (positive) square root: $\sqrt{25} = 5$. The negative square root is written as $-\sqrt{25} = -5$. Within real numbers, the square root of a negative number is undefined because no real number squared produces a negative value.

   What is a Cube Root?

   The cube root of a number is a value that, when multiplied by itself three times, produces the original number. The cube root of a number a is denoted by:

   $\sqrt[3]{a} = b$ because $b^3 = a$

   Perfect Cubes

   An integer whose cube root is an integer is called a perfect cube.

   • $\sqrt[3]{1} = 1$ because $1^3 = 1$
   • $\sqrt[3]{8} = 2$ because $2^3 = 8$
   • $\sqrt[3]{27} = 3$ because $3^3 = 27$
   • $\sqrt[3]{64} = 4$ because $4^3 = 64$

   Cube Roots of Negative Numbers

   Unlike square roots, cube roots of negative numbers are well-defined real numbers because an odd power of a negative number remains negative:

   • $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$
   • $\sqrt[3]{-27} = -3$ because $(-3)^3 = -27$
   • $\sqrt[3]{-125} = -5$ because $(-5)^3 = -125$

   Key Distinctions Between Square and Cube Roots

   • Index: A square root has an implied index of 2 ($\sqrt{x} = \sqrt[2]{x}$), while a cube root has an explicit index of 3 ($\sqrt[3]{x}$).
   • Domain: In real numbers, $\sqrt{x}$ requires $x \ge 0$, whereas $\sqrt[3]{x}$ is defined for all real values of $x$.

   Worked Examples

   Example 1: Evaluate $\sqrt{81} - \sqrt[3]{-64}$.

   Step 1: Compute $\sqrt{81} = 9$.
   Step 2: Compute $\sqrt[3]{-64} = -4$.
   Step 3: Subtract: $9 - (-4) = 9 + 4 = 13$.

   Example 2: Evaluate $\sqrt{\frac{49}{100}} + \sqrt[3]{\frac{1}{8}}$.

   Step 1: $\sqrt{\frac{49}{100}} = \frac{\sqrt{49}}{\sqrt{100}} = \frac{7}{10}$.
   Step 2: $\sqrt[3]{\frac{1}{8}} = \frac{\sqrt[3]{1}}{\sqrt[3]{8}} = \frac{1}{2} = \frac{5}{10}$.
   Step 3: Combine: $\frac{7}{10} + \frac{5}{10} = \frac{12}{10} = \frac{6}{5}$.