Raleigh Charter High School



Unit 9.1 – Radical Expressions and Graphs

[pic].

In the root, a is the radicand, n is the index, and the symbol is known as the radical sign.

Remember that numbers have 2 different roots. For example, 2 and -2 are both square roots of 4. They are also fourth roots of 16. We now introduce the concept of the principal root.

[pic]Root

If n is even and a is positive, then

[pic]is the principal [pic]root of a, and

[pic]is the negative [pic]root of a.

If n is even and a is negative, then

[pic]is not a real number.

If n is odd, then

there is exactly one [pic]root of a, written [pic].

(If n is even, then the two roots of a are often written [pic].)

Graphing Functions Defined with Radicals

[pic]

[pic]

[pic]

[pic]

Simplifying Square Roots Using Absolute Values

[pic]

If n is an even positive integer, [pic],

And if n is an odd positive integer, [pic].

Examples:

[pic]

Use a calculator to find roots.

Demonstrate how to use calculator to find roots other than square roots. Either store value in x and then use MATH menu or enter the index and then use MATH menu.

Unit 9.2 – Rational Exponents

Use the laws of exponents to resolve what this means:

[pic] but [pic] therefore [pic]

General rule:

If [pic] is a real number, then [pic].

Examples:

[pic]

Unit 9.3 – Simplifying Radical Expressions

Product Rule for Radicals

If [pic] and [pic]are real numbers and n is a natural number, then [pic]. (Remember that the product rule can only be used when the indexes are the same.)

Examples:

[pic]

Quotient Rule for Radicals

If [pic] and [pic]are real numbers and n is a natural number, then [pic].

Examples:

[pic]

Simplifying Radicals

When is a Radical considered simplified?

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 has a radical.

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

Example:

To simplify [pic], first check to see if the radicand is divisible by a perfect square (the square of a natural number) such as 4, 9, …. Choose the largest perfect square that divides in to 24, which is 4. Write 24 as the product of 4 and 6, then use the product rule.

[pic]

[pic]it may not be obvious that 108 is divisible by the perfect square 36. In such a case, factor the radicand into its prime factors to aid in identifying perfect squares.

[pic]

[pic]look for the largest perfect cube root that divides into 16.

[pic]

Examples:

[pic]

Examples using radicals with variables:

[pic]

[pic]

[pic]

Examples:

[pic]

Simplify Radicals by Using Smaller Indexes

Sometimes we can write a radical using rational exponents and then simplify the rational exponent to lowest terms. Then we write the answer as a radical.

[pic]

Examples:

[pic]

Multiplying Radicals with Different Indexes

Basically we rewrite the radical using rational exponents. Then we convert the rational exponents into new exponents with a common denominator. Rewrite as radicals and use the product rule.

[pic]

Example:

[pic]

Unit 9.4 – Adding and Subtracting Radical Expressions

Simplify radical expressions involving addition and subtraction

[pic]

This is exactly the same process as [pic].

Only radical expressions with the same index and the same radicand can be combined. Expressions such as [pic]cannot be simplified by combining terms.

Example:

[pic]

Examples:

[pic]

Example:

[pic]

Examples:

[pic]

Adding and Subtracting Radicals with Fractions

Example:

[pic]

Examples:

[pic]

Unit 9.5 – Multiplying and Dividing Radical Expressions

Multiply binomial expressions by using the FOIL method.

[pic]

(this result cannot be simplified further)

Examples:

[pic]

Rationalizing the denominator (with square roots)

This is the process of removing radicals from the denominator of a fraction.

Example:

[pic]

It can be easier if you simplify the radicals before rationalizing.

[pic]

Examples:

[pic]

Rationalizing denominators (with cube roots)

[pic] First use the quotient rule to simplify the numerator and denominator.

[pic]

To get a rational denominator, multiply the numerator and denominator by a number that will result in a perfect cube in the radicand of the denominator. In this case, since [pic], multiply the numerator and denominator by [pic].

[pic]

Examples:

[pic]

Rationalize denominators with binomials involving radicals

Ask them to recall what happens when you multiply [pic]: you get the difference of squares [pic].

We use the same process to simplify a denominator with radicals, such as [pic].

We call binomials such as [pic] and [pic] conjugates.

Unit 9.6 – Solving Equations with Radicals

An equation that includes one or more radical expressions with a variable is a radical equation.

[pic]

Power Rule for Solving Equations with Radicals

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.

BE CAREFUL: the power rule does not say that all solutions of the new equation are solutions of the original. You must check every solution in the original equation to be sure.

Examples:

[pic] check this answer: [pic]

Steps to solve an equation with radicals

1. Isolate the radical. Make sure that one radical term is alone on one side of the equation

2. Apply the power rule. Raise both sides of the equation to a power that is the same as the index of the radical.

3. Solve. Solve the resulting equation; if it still contains a radical, repeat Steps 1 & 2.

4. Check all potential solutions in the original equation.

Examples:

[pic] check your answer [pic]

3+3 obviously is not 0, so 2 is not a solution and the solution set is null.

[pic]

Remember to check your answers in the original equation. In this case -5 does not work in the original equation so the solution set is {0}.

[pic]

Check both answers in the original equation to find that only -1 works. So the solution set is {-1}.

How about powers greater than 2?

[pic]

Checking this answer in the original equation shows the solution set is {11}.

Examples:

[pic]

Unit 9.7 – Complex Numbers

We are going to talk about a new set of numbers. A set that includes the real numbers, as well as numbers that are even roots of negative numbers, such as [pic].

In order to do this, a new number was discovered or invented around the time of the Reformation (1500’s). It is called the ‘imaginary unit’ and is denoted by i. It is thought it was called “imaginary” because everyone ‘knew’ there was no use for such a number. It turns out that there are uses in electrical engineering and other fields.

The imaginary unit is defined as the number such that [pic] so [pic].

Using this we can define the square root of any negative number as

[pic]

Examples:

[pic]

Examples:

[pic]

Multiplying square roots of negative numbers

The product rule for radicals does not work because it only applies to positive radicands.

[pic]

Notice that the product rule does not work!

[pic]

Examples:

[pic] [pic] [pic]

Dividing square roots of imaginary numbers

This follows a similar process.

Examples:

[pic] [pic]

Examples:

[pic]

Complex Numbers

Complex numbers are of the form a + bi, where a and b are real numbers. The number a is called the real part and b is called the imaginary part. If a is 0 they are called pure imaginary numbers.

Add and subtract complex numbers

We use the commutative, associative, and distributive properties to add and subtract complex numbers.

To add, we add the real parts and add the imaginary parts.

Examples:

[pic] [pic]

To subtract, we subtract the real parts and subtract the imaginary parts.

Examples:

[pic] [pic]

[pic]

Examples:

[pic] [pic]

Multiplying complex numbers

To multiply complex numbers, treat them like polynomials and FOIL them.

Examples:

[pic] [pic]

Examples:

[pic]

Dividing complex numbers

The quotient of two complex numbers should be a complex number. We do need to eliminate the i from the denominator. To do this we use the conjugate of the denominator.

Examples:

[pic]

Examples:

[pic]

Finding powers of i

The powers of i rotate through 4 numbers: [pic]

So, for example [pic] which suggests a method to simplify large powers of i:

Divide the exponent by 4, throw away the answer and just keep the remainder as the exponent of i.

[pic]

Examples:

[pic]

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