Find the least common denominator for each set of rational expressions. Write each expression in

step 1 In this problem, we want to find our least common denominator. The way that we can approach this

Find the least common denominator for each set of rational...

Key Concepts

Rational Expressions

Rational expressions are fractions that involve polynomials in both the numerator and the denominator. Understanding how to manipulate these expressions, including operations like addition and subtraction, requires expressing them in a common format or denominator.

Least Common Denominator (LCD)

The least common denominator is the smallest expression that can serve as a common denominator for a set of rational expressions. It incorporates all factors from the individual denominators at their highest powers, ensuring that each original denominator divides the LCD.

Factorization

Factorization involves decomposing expressions into products of simpler factors. In the context of finding an LCD, factorizing each denominator is essential because it helps identify the requisite factors and their exponents needed to form the least common multiple.

Equivalent Forms of Rational Expressions

Writing expressions in terms of the LCD involves creating equivalent forms by multiplying the numerator and denominator by the necessary factors. This process is key for aligning denominators, which allows the expressions to be combined or compared effectively.

Transcript

00:01 In this problem, we want to find our least common denominator.

00:03 The way that we can approach this if you don't know how to find it is by taking a look essentially at the breakdown of these.

00:09 For example, 15 can be rewritten as its prime factorization of 3 times 5.

00:14 I have 2bs for b squared and then i have a c.

00:18 25 is 5 times 5.

00:21 I have 1a and here i have myself 3 c's.

00:24 So now to come with the least common denominator using what i see here, as long as we have a monominy like this, you can see what's on the left that's not on the right and vice versa.

00:34 For example, i have a 3 times 5 on the left, but on the right i have 5 times 5.

00:39 So i know i'm going to have to bring in a 3.

00:42 Continue with that idea.

00:43 I have a 5, a 5 at a 3 now.

00:45 On the left, i'm going to need another 5 to get the 2 5s from the right -hand side.

00:50 So now i'm looking at the variables.

00:52 On the right, i have an a, where on the left i do not.

00:55 So i'm going to bring an a over there.

00:57 I have 2bs on the left, where on the right i don't have any.

01:01 So i know i'm going to have to bring in two bs now.

01:03 Finally, i have three cs on the right and one on the left, so i know i have to bring in two cs.

01:09 So just comparing now, i see that i have a three and two fives, one a, two bs, and three cs.

01:17 So this has told me that now we have our least common denominator.

01:21 We can multiply those all together, but let's go ahead and change them using what i have here...

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