Intermediate Algebra Chapter 7

Section 7.1 Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b 2 = a. In order to find a square root of a, you need a # that, when squared, equals a.

The principal (nonnegative) square root is noted as The negative square root is noted as

Radical expression is an expression containing a radical sign. Radicand is the expression under a radical sign. Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

Example

Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers. Otherwise, leave them in radical form. Do not convert to an approximation unless requested to do so.

Radicands might also contain variables and powers of variables. Example Simplify. Assume that all variables represent positive numbers.

The cube root of a real number a Note: a is not restricted to non-negative numbers for cubes. The cube root of a negative number is a negative number.

Example

Other roots can be found, as well. The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number when a is negative.

Simplify the following. Assume that all variables represent positive numbers. Example

If the index of the root is even, then the notation represents a positive number. But we may not know whether the variable a is a positive or negative value. Since the positive square root must indeed be positive, we might have to use absolute value signs to guarantee the answer is positive.

If n is an even positive integer, then If n is an odd positive integer, then

Simplify the following. Example If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign.

Example Simplify the following. Since the index is odd, we don’t have to force the negative root to be a negative number. If a or b is negative (and thus changes the sign of the answer), that’s okay.

Since every value of x that is substituted into the equation produces a unique value of y, the root relation actually represents a function. The domain of the root function when the index is even, is all nonnegative numbers. The domain of the root function when the index is odd, is the set of all real numbers.

We have previously worked with graphing basic forms of functions so that you have some familiarity with their general shape. You should have a basic familiarity with root functions, as well.

Example x y x y (0, 0) (4, 2) (1, 1) Graph 6 2 (2, ) (6, )

Example x y x y (0, 0) (1, 1) Graph 28 4 (4, ) (8, 2) (-1, -1) (-4, )(-8, -2)