Simplifying Fractions

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來自 University of California, Irvine 的課程

Pre-Calculus: Functions

350 個評分

University of California, Irvine

Pre-Calculus: Functions

350 個評分

This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Solve linear and quadratic equations 2. Solve some classes of rational and radical equations 3. Graph polynomial, rational, piece-wise, exponential and logarithmic functions 4. Find integer roots of polynomial equations 5. Solve exponential and logarithm equations 6. Understand the inverse relations between exponential and logarithm equations 7. Compute values of exponential and logarithm expressions using basic properties

從本節課中

Rational and Radical Equations.

We begin this module by practicing manipulations of rational expressions using basic properties. We then utilize these manipulations to solve rational and radical equations.

Simplifying Fractions 5:42

Simplifying a Ratio of Polynomials 3:31

Adding and Subtracting Rational Expressions 8:03

Multiplying Rational Expressions 3:17

Dividing Rational Expressions 4:01

與講師見面

Dr. Sarah Eichhorn
Associate Dean, Distance Learning & Tenured Lecturer
Dr. Rachel Cohen Lehman

[SOUND] Let's look at Simplifying Fractions.

For example, let's simplify this expression here.

Now, these types of fractions where the numerator or the denominator or both

contain a fraction are often referred to as either Complex Fractions or Compound

Fractions. However, don't confuse the word complex

here with the complex number. System.

Now remember, when we're dividing 2 fractions, we multiply the numerator by

the reciprocal of the denominator. That is, we flip the bottom fraction, and

multiply. And when we multiply fractions, we

multiply the numerators. And we multiply the denominators.

And now this x^2 - 9 will factor. It's a difference of two squares, which

gives us 5 * 3 + x / 15x * and this factors into x + 3 * x - 3.

Now, x + 3 and 3 + x will cancel, and we're assuming here of course that x is

not equal negative 3. And 5 goes into 15, 3 times, so we'll

have a 3 left here in the denominator. And therefore, our answer is 1 / 3x

(x-3). Alright, let's see another example.

Let's simplify this fraction. Now, there are different approaches here.

For example, we could work with the numerator separately and then work with

the denominator separately. However the method that students seem to

like best when working with this type of fraction, is to first eliminate.

These denominators here and we can do that by multiplying both the numerator

and denominator by the least common multiple of the denominators, which in

this case is their product x times y. So lets do that.

We'll multiply the numerator by x times y, as well as the denominator.

Again when multiplying fractions, we multiply the numerators.

And we multiply the denominators. And now we'll distribute this xy to these

three terms, as well as distribute this xy to these two terms, which gives us.

x / y * xy + -2 * xy + y / x * xy, all / x / y.

* xy + -y/x (xy). Now the y's will cancel here as well as

the x's. And the samw thing in the denominator,

the y's will cancel. As well as the x's and we are assuming

here of course that x and y both are non-zero.

and what are we left with? We have x * x which is x^2 and then we have - 2xy and

then + y * y which is y^2. Divided by x * x, which is x^2, and minus

y * y, which is y^2. And look, we've eliminated those original

denominators, and now we can factor. Namely, this is equal to ( x - y ) ^ 2 /,

and the denominator is the difference of 2 squares which factors into x - y * x +

y. And now we can cancel an x - y, from both

the denominator and the numerator, which will leave us with 1 in the numerator.

And we are assuming here of course that x - y is non zero, or that x does not equal

y. Which leaves us with our answer of (x -

y) / (x + y), which doesn't look quite as complex, does it? And this is how we

simplify these types of fractions. Thank you and we'll see you next time.

[SOUND]

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