Finding the square root of a number and squaring a number are inverse operations.
To find the square root of a number n, you must find a number whose square is n. For example, a square root of 49 is 7 since 72 = 49. Likewise, (–7)2 = 49, so –7 is also a square root of 49.
Definition of Square Root
For any real numbers a and b, if a2 = b, then a is a square root of b. NOTE: Every positive real number has two real number square roots. The number 0 has just one square root, 0 itself. Negative numbers do not have real number square roots.
Understanding the symbols
This symbol represents the principal square root of a.
The principal square root of a non-negative number is its nonnegative square root.
For example:
In other words,
To name the negative square root of a we use this symbol.
For example:
To indicate both square roots, use the plus/minus sign which indicates positive or negative.
For example:
Negative numbers do not have real number square roots.
For example:
Terminology
The symbol
is called a radical sign. An expression written with a radical sign is called a radical expression. The expression written under the radical sign is called the radicand.
examples:
Simplify completely. Do not use a calculator.
NOTE:
Since m could be any real number value, positive or negative, and the symbol used in the principal square root, we should indicate the absolute value of m to ensure the result is non-negative.
NOTE: A perfect square trinomial can be factored to be the product of a binomial squared. Absolute value bars are needed since this is the principal square root.
Cube Roots
Definition of Cube Root
The number c is the cube root of a if its third power is a, that is, c3 = a. Every real number has exactly one cube root in the real number system.
For example: 2 is the cube root of 8 because 23 = 8, or.
–3 is the cube root of –27 because (–3)3 = –27, or
NOTE:
No absolute value signs are needed when finding cube roots, because a real number has just one cube root. The cube root of a positive number is positive. The cube root of a negative number is negative.
nth Roots
For any real numbers a and b, and any positive integer n, if an = b, then a is the nth root of b.
example:
Whenever the number n in is an odd number, we say we are taking an odd root. When the index is an even number, we say that we are taking an even root. The number n is called the index. When the index is 2 we do not write it.
When we take any odd root of a number, we find that there is just one answer. If the number is positive, the root is positive. If the number is negative, the root is negative.
Every positive real number has two nth roots when n is even. One of these roots is positive and one is negative. Negative real numbers do not have nth roots when n is even.
Absolute value signs are never needed when finding odd roots. When finding even nth roots, absolute value signs are sometimes necessary, as with square roots.
examples:
odd index – one answer only, no absolute value bars needed.
even index – two nth roots are possible, we only want the non-negative root. Absolute value bars are needed to ensure y is non-negative.
odd index – one answer only, a negative constant, no absolute value bars needed since index is odd.
