This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: understand the concept of square root, be able to distinguish between the principal and secondary square roots of a number, be able to relate square roots and meaningful expressions and to simplify a square root expression.
Overview
- Square Roots
- Principal and Secondary Square Roots
- Meaningful Expressions
- Simplifying Square Roots
Square roots
When we studied exponents in Section
[link] , we noted that
and
We can see that 16 is the square of both 4 and
. Since 16 comes from squaring
4 or
, 4 and
are called the
square roots of 16. Thus 16 has two square roots, 4 and
. Notice that these two square roots are opposites of each other.
We can say that
Square root
The square root of a positive number
is a number such that when it is squared the number
results.
Every positive number has two square roots, one positive square root and one negative square root. Furthermore, the two square roots of a positive number are opposites of each other. The square root of 0 is 0.
Sample set a
The two square roots of 49 are 7 and −7 since
The two square roots of
are
and
since
Practice set a
Name both square roots of each of the following numbers.
Principal and secondary square roots
There is a notation for distinguishing the positive square root of a number
from the negative square root of
.
Principal square root:
If
is a positive real number, then
represents the positive square root of
. The positive square root of a number is called the
principal square root of the number.
Secondary square root:
represents the negative square root of
. The negative square root of a number is called the
secondary square root of the number.
indicates the secondary square root of
.
Radical sign, radicand, and radical
In the expression
is called a
radical sign .
is called the
radicand .
is called a
radical .
The horizontal bar that appears attached to the radical sign,
, is a grouping symbol that specifies the radicand.
Because
and
are the two square root of
,
Sample set b
Write the principal and secondary square roots of each number.
Use a calculator to obtain a decimal approximation for the two square roots of 34. Round to two decimal places.
Notice that the square root symbol on the calculator is
. This means, of course, that a calculator will produce only the positive square root. We must supply the negative square root ourselves.
Note: The symbol ≈ means "approximately equal to."
