Roots and Radicals: Square Roots, nth Roots, and Imaginary Numbers

Roots and Radicals

Square Roots

The square root of a number is the reverse operation of squaring. For any positive real number, there are always two square roots: a positive (principal) root and a negative root.

• Definition: The square root of a number $a$ is a number $x$ such that $x^2 = a$.

• Notation: $\sqrt{a}$ denotes the principal (positive) square root.

• Radicand: The term inside the radical symbol ($\sqrt{\phantom{a}}$) is called the radicand.

• Radical Symbol: $\sqrt{\phantom{a}}$

• Roots of Positive Numbers: $\sqrt{9} = 3$ and $-\sqrt{9} = -3$

• Roots of Negative Numbers: $\sqrt{-9}$ is imaginary because no real number squared gives a negative result.

Warning: Negative numbers cannot be square rooted in the set of real numbers; their roots are imaginary.

• Memory Tool: Negatives outside the radical: okay. Negatives inside the radical: imaginary.

Examples

• $\sqrt{36} = 6$

• $-\sqrt{36} = -6$

• $\sqrt{-36}$ is imaginary

Practice Problems

• $5 = \sqrt{\frac{1}{4}}$

• $\sqrt{(-5)^2} = 7$

• Additional info: The second example should be $\sqrt{(-5)^2} = 5$; the answer "7" is likely a typo.

Even vs. Odd Roots

The nth root of a number is the reverse operation of raising a number to the nth power. The index $n$ is written at the top-left of the radical symbol: $\sqrt[n]{a}$.

• nth Root Formula: $a^{1/n} = \sqrt[n]{a}$

• Even Index (n = 2, 4, 6...):

  • Even roots of positive numbers have two roots: positive and negative.

  • Even roots of negative numbers are imaginary.

  • Examples:

    • $2^2 = 4$ so $\sqrt{4} = 2$

    • $(-2)^2 = 4$ so $\sqrt{4} = -2$

    • $\sqrt{-4}$ is imaginary

• Odd Index (n = 3, 5, 7...):

  • Odd roots of positive and negative numbers exist and have the same sign as the radicand.

  • Examples:

    • $2^3 = 8$ so $\sqrt[3]{8} = 2$

    • $(-2)^3 = -8$ so $\sqrt[3]{-8} = -2$

Table: Comparison of Even and Odd Roots

Index	Radicand Positive	Radicand Negative	Number of Real Roots	Imaginary?
Even (2, 4, 6...)	Yes	No	2 (positive & negative)	Yes (if radicand is negative)
Odd (3, 5, 7...)	Yes	Yes	1	No

Examples

• $\sqrt[3]{8} = 2$

• $\sqrt[3]{-27} = -3$

• $\sqrt[4]{16} = 2$

• $\sqrt[4]{-16}$ is imaginary

Perfect Powers Reference Table

Squares	Cubes	Fourth Powers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144	1, 8, 27, 64, 125, 216, 343, 512, 729, 1000	1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000

Additional Info

• If a term in $n$th root has an exponent equal to the index, they cancel out, leaving just the radicand. For example, $\sqrt[3]{-21}$ is imaginary, $\sqrt[5]{-5} = -1$.