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Lgebra II. module 2docx

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LGEBRA II : RATIONAL, COMPLEX, AND POLYNOMIALS : 02 RATIONAL EXPONENTS

Rational Exponents to Radical Expressions

Will the same process apply when the numerators are greater than 1? Take a look at the rational

exponent of. This is the same as.

If these three factors were rewritten as radicals, the result would be. But remember, when radicals of the same root are multiplied, the terms inside the radical are

multiplied, resulting in. Consequently,

=

Example 1

Rewrite in radical form.

Example 2

Rewrite in radical form.

Notice that only the variable is being raised to the power of. The 6 is not. If the 6 were being

raised to that power, the expression would be written as. Consequently, only the variable will be rewritten as a radicand.

Radical Expressions to Rational Exponents

Not only can rational exponents be written as radical expressions, the reverse is also true. Radical expressions may be written as rational exponents.

Take a look at. If this radicand were to be expanded, the result would be. This

expanded form written with rational exponents would look like. Consequently, these

rational exponents can be added to result in.

Therefore,.

Example 1

Rewrite as a rational exponent.

Example 2

Rewrite with a rational exponent.

In this case, everything inside the radical will be raised to the rational exponent. As a result, parentheses will be needed. When there is no visible exponent underneath the radical, an understood 1 is in its place. Remember, this exponent becomes the numerator of the rational exponent and the root of the radical becomes the denominator.

Example 1

You already know how to multiply and divide radicals of the same root (or index). The radicands are multiplied together while the index remains the same.

But how do you multiply radicals that have different roots, like? You could attempt to multiply the radicands together but then what would the index be? Believe it or not, you have all the tools you need to figure this out. Check out the Product of a Power Property.

To multiply powers of the same base, add the exponents.

Example:

Now, let’s see if you can find the product.

Example 2

Express the product of in radical form.

Quotient of a Power Property

To divide powers of the same base, subtract the exponents.

Example:

Example 1

Express the quotient of as a radical.

Example 2

Express the product of as a rational exponent.

Negative Rational Exponents

Raising a Base to a Negative Rational Exponent

Recall that the inverse of positive is negative, and the inverse operation of multiplication is division. When the exponent is positive, it tells you to multiply the base the number of times indicated by the exponent.

n times

Inversely, a negative exponent tells you to divide by the base the number of times indicated by the exponent.

n times

Simplify. Express the answer using a rational exponent.

Simplify.

ALGEBRA II : RATIONAL, COMPLEX, AND POLYNOMIALS : 02 SOLVING RADICAL

EQUATIONS

Defining Radical Equations

Although you haven't learned how to solve radical equations yet, you can still see one in action. The time it takes for a pendulum to complete one full swing—to one side and back again—can be found using the following radical equation:

If you will recall from prior geometry courses, the formula for the circumference of a circle is 2 r. In the case of the formula given here, the r for radius in the circumference formula is simply replaced by

the radical.

Identifying Radical Equations

This is a radical equation:

This is not a radical equation:

What is the difference? In the first equation, the variable is inside the radical. In the second equation, the variable is outside the radical. Therefore, a radical equation can be defined as an equation containing a radical where a variable is, or is a part of, the radicand.

To ensure your understanding of identifying radical equations, complete the following exercise by dragging and dropping each equation into the appropriate column.

What operation would you use to solve an equation such as? The opposite of taking the square root of a variable is raising that variable to the second power. So square both sides of the equation.

x = 16

How does squaring isolate the variable? Recall, , and two identical factors inside the radical simplify to a single factor outside of the radical.

1.

2. Check Your Work!

Practice

1.

2.

3.

4.

5.

1.

2.

3.

Examples: , , 3... (Any real number that does not fit into any of the other four categories)  To better understand this Real Number System, take a look at the following diagram that represents the relationship between each category.



Classify each number as a Natural, Non-Natural, Whole, Non-Whole, Integer, Non-Integer, Rational and Irrational: 5, 0,−3, 2/3,−1/7, 2..., π, √

Natural number: 5, Non-Natural numbers: 0,−3, 2/3,−1/7, 2..., π, √3; Whole number: 5, 0 Non- Whole numbers: −3, 2/3,−1/7, 2..., π, √3; Integers: 5, 0,−3 Non-Integers: 2/3,−1/7, 2..., π, √3; Rational numbers: 5, 0,−3, 2/3,−1/7, Irrational Numbers: 2..., π, √

Imaginary Numbers

What pair(s) of identical factors will produce −1 when multiplied? 1 times 1 doesn’t work since it equals positive 1. The same is true of −1 times −1 which also equals positive 1. 1 times −1 does equal −1, but these are not identical factors.

To solve this problem, the concept of the imaginary number i was invented. The imaginary number was defined to be

With this definition, the square root of negative radicands, in addition to positive radicands, may be simplified.

Example 1

Simplify.

i 2 :

i 3 :

i 4 :

Based on this information, to what will i 5 simplify?

Any complex number can be written in the form a + bi, where a represents the real number term and bi represents the imaginary number term. In each of the complex numbers below, drag and drop the real number into the real number column and the imaginary number into the imaginary number column.

Conjugates of Complex Numbers

When you mastered radicals, you identified the conjugate of a radical as a binomial whose terms are identical to another binomial, but with the opposite sign separating those terms. For example, the

conjugate of is. The same concept can be applied to complex numbers. The conjugate of 7 + 10i is 7 − 10i. All you are doing is finding the opposite sign of the imaginary part of the number.

Practice

Find the conjugate of 2+3i.

ALGEBRA II : RATIONAL, COMPLEX, AND POLYNOMIALS : 02 OPERATIONS OF COMPLEX

NUMBERS

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