Overview
The root of any number is specified by the radical symbol (√). It is the inverse operation to raising a number by an exponent. Roots may be rational numbers or irrational numbers.
What Is the Square Root?
The square of any number is the number multiplied by itself. For example, 62 equals 36. The inverse of that operation is taking the square root, specified with the radical symbol without a number above it. If 36 equals 62, then √36 equals 6. Similarly, the square root of 16 is 4, because 4 times 4 equals 16.
What Is the Principal Root?
However, there are 2 square roots for 36, both 6 and -6, because 62 is equal to 36, and (-6)2 is also equal to 36. However, the positive root of a number is referred to as the principal square root of the number. When both the positive and negative roots of a number are desired, the ± is used before the radical sign, and when just the negative square root is desired the symbol -√ is used.
What About Negative Numbers?
The square root of a negative number is not a real number. There is no real number that multiplied by itself that will equal a negative number. For example, √-25 does not equal -5, because -5 times -5 is not -25, but 25. Algebraically speaking, for any real number √a2 to equal a, a must not be negative. It can be 0, because the square root of 02 or 0 times 0 is 0. Therefore, the square root of x4 is x2 as long as x is positive or zero, in order for the solution to be real.
What About Rational and Irrational Numbers?
Many numbers are perfect squares and have rational square roots. For example, the square root of 16 is exactly 4 because 4 times itself is 16. The square root of 25 is exactly 5, because 5 times 5 is 25. What about the square root of 20? It will be somewhere between 4 and 5, but can only be approximated, as an irrational number. If the sides of a right triangle can be expressed in the Pythagorean theorem by a2 + b2 =c2, then the length of any side can either be a rational or an irrational number. For example, suppose one side measures 6 and one side measures 4. Then c2 will equal 62 +42, or 36 +16 =52. Side c will measure √52, which is an irrational number between 7 and 8.
Interested in math tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy’s is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Roanoke, VA visit: Tutoring in Roanoke, VA