Principal Square Root

Definition

The principal square root of a number is the positive square root of that number. It is the value that, when multiplied by itself, gives the original number. The principal square root is a fundamental concept in solving equations with square roots.

5 Must Know Facts For Your Next Test

1. The principal square root of a number is always the positive square root, even if the number is negative.
2. When solving equations with square roots, the principal square root must be considered to find the valid solutions.
3. The principal square root of a perfect square (a number that is the square of an integer) is the integer itself.
4. The principal square root of a negative number is not a real number, but rather an imaginary number.
5. The principal square root is a key concept in understanding and solving equations involving square roots, such as $\sqrt{x} = 4$.

Review Questions

• Explain the relationship between the principal square root and the square of a number.
  • The principal square root of a number is the positive value that, when multiplied by itself, gives the original number. For example, the principal square root of 9 is 3, because 3 × 3 = 9. This means that the principal square root and the square of a number are inverse operations, where the square root undoes the squaring operation.
• Describe the properties of the principal square root when dealing with negative numbers.
  • The principal square root of a negative number is not a real number, but rather an imaginary number. This is because the square of any real number, whether positive or negative, will always be a non-negative number. Therefore, the principal square root of a negative number is not defined in the real number system, and special mathematical concepts, such as complex numbers, are required to work with square roots of negative numbers.
• How does the principal square root relate to solving equations with square roots, such as $\sqrt{x} = 4$?
  • When solving equations with square roots, the principal square root must be considered to find the valid solutions. For example, to solve the equation $\sqrt{x} = 4$, we need to recognize that the principal square root is the positive square root. Therefore, the solution to this equation is $x = 16$, because $\sqrt{16} = 4$. The principal square root ensures that the solutions obtained are consistent with the original equation and its context.

Related terms

Square Root: The square root of a number is the value that, when multiplied by itself, gives the original number. The principal square root is the positive square root.

Radical Notation: The symbol $\sqrt{}$ is used to represent the square root of a number, with the number inside the radical sign.

Exponents: The exponent $1/2$ is used to represent the square root of a number, as in $x^{1/2}$, which is equivalent to $\sqrt{x}$.