Basic Technical Mathematics, 11th Edition

1.6 Roots and Radicals

• Principal nth Root • Simplifying Radicals • Using a Calculator • Imaginary Numbers

At times, we have to find the square root of a number, or maybe some other root of a number, such as a cube root. This means we must find a number that when squared, or cubed, and so on equals some given number. For example, to find the square root of 9, we must find a number that when squared equals 9. In this case, either 3 or $\hspace{0.17em}-\hspace{0.17em}3$ is an answer. Therefore, either 3 or $\hspace{0.17em}-\hspace{0.17em}3$ is a square root of 9 since $3^2\hspace{0.17em}=\hspace{0.17em}9$ and ${(\hspace{0.17em}-\hspace{0.17em}3)}^2\hspace{0.17em}=\hspace{0.17em}9.$

To have a general notation for the square root and have it represent one number, we define the principal square root of a to be positive if a is positive and represent it by $\sqrt{a}\hspace{0.17em}.\hspace{0.17em}$ This means $\sqrt{9}\hspace{0.17em}=\hspace{0.17em}3$ and not $\hspace{0.17em}-\hspace{0.17em}3.$