# Roots, Radicals, and Root Functions

**Conditions for a Simplified Radical**

1. The radicand has no factor raised to a power greater
than or equal to the index.

2. The radicand has no fractions.

3. No denominator contains a radical.

4. Exponents in the radicand and the index of the radical have no common
factors, except 1.

**Examples: **Fully simplify the given radical
expressions.

**Simplifying Radicals**

• Recall that to simplify a square root we look for a number multiplied by
itself twice to give us the radicand

• A similar idea holds for all values of the index

• If we cannot find a value multiplied by itself twice to give us the radicand
in a square root, we then look for square multiples of the radicand

• If the index was three, we would be looking for cube multiples of the
radicands

**Examples:** Fully simplify the given radical expressions.

**The Pythagorean Relation**

• If c is the length of the longest side of a right triangle and a and b are the
lengths of the shorter sides, then.

• The longest side is the hypotenuse, and the two shorter sides are the legs of
the triangle.

• The hypotenuse if the side opposite the right angle.

**Examples: **Find the length of the unknown side in the given right triangle.

**Distance Formula**

• To determine the distance between two points,
and on
a coordinate plane

•

**Examples: **Find the distance between the pair of given points.

a) (2, -1) & (5, 3)

**Section 10.4: Adding and Subtracting Radical Expressions**

• When adding and subtracting radical expressions we must have the same index
and radicand

• If the radicands are not the same then we try to simplify the radicands so
they are the same

**Examples: **Add or subtract to simplify each radical expression. Assume that all
variables represent positive real numbers.

**Section 10.5: Multiplying and Dividing Radical Expressions**

• When multiplying radical expressions we can use the distributive property

• When multiplying radical expressions with binomials we FOIL

• Recall, when multiplying radical expressions we can only multiply radicals
with the same index

**Examples:** Multiply the radical expressions.

**Rationalizing Denominators**

• Recall that a radical expression is not fully simplified if radicals are
present in the denominator of a fraction

• We use the method of rationalizing denominators to remove the radical from the
denominator of a fraction

• Note,

**Examples: **Rationalize each denominator.

**Rationalizing a Binomial Denominator**

• Whenever a radical expression has a sum or difference with square root
radicals in the denominator, rationalize the denominator by multiplying both the
numerator and the denominator by the conjugate of the denominator

• Conjugates have the same terms with opposite signs;

**Examples:** rational each denominator.

**Writing Radical Quotients in Lowest Terms**

• Factor the numerator and the denominator

• Divide out any common factors

**Examples: **Write each quotient in lowest terms.

**Section 10.6: Solving Equations with Radicals**

• We now look at how to solve equations with radical expressions

• We use the following fact to solve these types of equations

-If both sides of an equation are raised to the same power, all solutions of the
original equation are also solutions of the new equation

**Solving an Equation with Radicals**

1. Isolate the radical

2. Apply the power rule

3. Solve the resulting equation

4. Check all solutions in the original equation

**Examples: **Solve the following equations.

**Section 10.7: Complex Numbers**

• Recall that the square root of a negative number is a non-real number

• A non-real number is also called a complex number

• We use complex numbers to define square roots of negative numbers

• The imaginary unit i is used to define any square
root of a negative real number 1 and

• For any positive real number b,

**Examples:** Write each number as a product of a real number and i

**Multiplying and Dividing Square Roots of Negative Numbers**

• The product rule and quotient rule only applies to nonnegative radicands

• We must change to the form
before performing multiplication

• Recall that, and

**Examples:** Multiply or divide as indicated.

**Complex Number**

• If a and b are real numbers, then any number of the form

Is called a complex number

• In the complex number, then number a is the real part and is the imaginary
part bi

**Adding and Subtracting Complex Numbers**

• Combine the real parts

• Combine the imaginary parts

**Examples: **Add or subtract the given expressions as indicated.

**Multiplying Complex Numbers**

• We use the distributive property

• FOIL binomials

• Recall that and

**Examples:** Multiply the given expressions. Give a fully simplified result.

**Dividing Complex Numbers**

• Multiply the numerator and denominator by the conjugate of the denominator

• Recall that conjugates are of the form

• We multiply by the conjugate to rationalize the denominator

**Examples:** Find each quotient. State the conjugate that will be used to
rationalize the denominator.