Solving Radical Equations

8.1 Definitions and Roots

▪ Need To Know
▪ How to find the root of a number
▪ Categorizing roots
▪ How to find the root of an expression
▪ How to solve application problems
involving roots

Idea of Square Roots

In mathematics, once we learn an operation,
we also learn the reverse of that operation.

For real numbers x and y,
If y = x² then x = ___

Don’t Forget
Radicand
Radical Sign
Radical

Definitions and Roots

Definition – If x is any positive real number, then
is the square root of x and
is the square root of x.

Examples:
Name the two square roots of 81.

Find the root of

Simplify

Evaluate

Recall Number Sets

▪ Rational Numbers
are the numbers of
the form a/b where
b is not zero.

▪ Rational Numbers is
the set of all fractions

Recall Number Sets

▪ Irrational Numbers
are the numbers
that can not be
written as fractions

▪ Real Numbers are
the collection of all
Rational and
Irrational Numbers.

Categorize Roots

Square Roots & Absolute Value

Consider and

Both come out positive.

Recall: |-7| = 7 and |7| = 7.

For all real numbers A, ___________________

Examples:

Application

The speed of sound in feet per second, V, traveling
through air with a temperature of t is given by the
formula below. Find the speed of sound when the
temperature is 5° C.

8.1 Conclusion

▪ The square root is the number that
“undoes” the square.
▪ Square roots can be positive or negative.
▪ The square root of zero is zero.
▪ The square root of a negative number is
not a Real Number
▪ A square root results in a
Rational, Irrational or Non-Real Number

8.2 Multiply and Simplify Radicals

▪ Need To Know

▪ Multiplying of radicals
▪ Simplifying radical expressions
1. With numbers
2. With variables

Multiplication Property of Radicals

▪ If A and B are real numbers (> 0), then
______________________________

Simplify Radical Expressions

Simplify Radical Expressions

▪ Simplify:

Perfect Square

Simplify Radical Expressions

▪ Simplify:

Square roots
undo squares

Practice Simplifying Radicals
▪ Simplify:

end

8.3 Properties of Radicals

▪ Need To Know
▪ Quotient Rule for Square Roots
▪ Simplifying radical expressions
1. With Fractions
2. By Rationalizing the Denominator

Quotient Rule for Square Roots
▪ If A and B are real numbers (B ≠ 0),
then
__________________________________

Simplify Radical Expressions
▪ Simplify:

The Idea of a Simplified Radical

Which fraction is the simplest?

Rationalizing Denominators

▪ Goal: Change the fraction to make the
denominator come out “nice”.

Guidelines for Simplification

1. Remove ____________________________
2. Remove ____________________________
3. Remove ____________________________

▪ Examples:

Guidelines for Simplification

1. Remove fractions from radicals
2. Remove perfects for radicals
3. Remove radicals from denominators

▪ Examples:

end