Square Roots, Cube Roots and Higher Roots

Recall when raising a number to a power n, where n is an integer greater than 1, we multiply the number by itself n times.

For example, 43 = 4 ∙ 4 ∙ 4. Now suppose we want to know what number multiplied by itself 2 times equals. Problems of this kind can be represented using radicals. A radical symbol √ is used to show the square root, or principal square root of a number or expression that appears under the radical symbol. Recall that the square root is defined as a number or expression multiplied by itself twice to equal the number or expression under the radical symbol, known as the radicand.

For example, if we want to know what number multiplied by itself 2 times equals 169, we can set this up with the radical symbol as follows:

√169, read as “square root of 169”. The answer to this is 13.
- √169, read as “negative square root of 169”. The answer to this is -13.

√0.09 = 0.3 and -0.3 since (0.3)2 and (-0.3)2 equals 0.09. The principal square root is 0.3. Another way to simplify this is to change √0.09 to √(9/100) and simplify to 3/10.

√(25/49) = 5/7 and -5/7 since (5/7)2 and (-5/7)2. The principal square root is 5/7.

• Note that a square root also has a negative value since a negative times a negative equals a positive, but we will deal with only the principal square root unless otherwise noted.
• Note that you can also simplify the square root of a fraction by taking the square root of the numerator and then the square root of the denominator instead of the square root of the fraction as a whole. In the previous example, you can take the square root of 25 first, then the square root of 49.
• Note that the square root of many positive integers are not whole numbers or rational numbers. For example, √19 can be found on a calculator or by leaving the answer as √19.

Sometimes we have to find the square root of a number that is not a perfect square. In these cases, we break down the radicand into factors, one of which is a perfect square.

Examples: Find each square root.
1. √68

First, find factors of 68.
Since 68 is even, we can divide it by 2. Therefore, 68 = 2 ∙ 34. Notice 34 is also divisible by 2, therefore 34 = 2 ∙ 17.

So 68 is factored into 2 ∙ 2 ∙ 17. Notice that 17 is prime and cannot be factored further and 2 ∙ 2 = 4, which is a perfect square. Therefore √68 = √4 ∙ √17 = 2√17.

2. √108
First, find factors of 108.

Since 108 is even, we can divide it by 2. Therefore 108 = 2 ∙ 54. Notice 54 is also divisible by 2, therefore 54 = 2 ∙ 27.

Next, we know that 27 = 3 ∙ 3 ∙ 3.
The factors of 108 are 2 ∙ 2 ∙ 3 ∙ 3 ∙ 3. Notice 2 ∙ 2 = 4, which is a perfect square and 3 ∙ 3 = 9, which is also a perfect square. Therefore, 108 = 4 ∙ 9 ∙ 3 and √108 = √4 ∙ √9 ∙ √3 = 2 ∙ 3 ∙ √3 = 6√3.

• Note that 108 = 36 ∙ 3 and 36 is a perfect square. But if you can't see right away that 3 is a factor of 108, you can break down by dividing 108 by 2 first and then simplify further at that point. It's easy to determine that 108 is divisible by 3. If the sum of the digits of a number are divisible by 3, the number is divisible by 3.