Intro to Radicals

Radicals: expressions in the form n√(a); means “the nth root of a“

Radical symbol: the symbol used to represent a radical; √

Index: the number of time the radicand should be divided by itself; n

Radicand: the number under the radical symbol; a

n√(a)

= a number

such that

(the number)n

= a

√(25)

=5

because

5²

=25

∛(8)

=2

because

2³

=8

∜(81)

=3

because

3⁴

=81

The index, n, must be a positive integer.

Restrictions on Radicands

Ex: √(-25) = a number only if (the number)2 = -25

There is no number that will satisfy the above condition; √(-25) is not a real number.

When the index of a radical is even, the radicand must be greater than or equal to zero and the radical must be positive.

Ex: ∛(-8) = a number only if (the number)3 = -8

The number that satisfies the above condition ∛(-8) is the negative number -2

When the index of a radical is odd, the radicand could be any real number.

  • If the radicand is positive, then the radical is postive

  • If the radicand is negative, then the radical is negative

Ex: For what values of x is √(7-3x) a real value?

7-3x ≥ 0 → -3x/3 ≥ -7/-3 → x ≤ 7/3

Ex: For what values of x is √(x2) a real value?

x∑R; x2 will always equal a positive value.