Identify the least common denominator of each group of rational expression, and rewrite each as

Identify the least common denominator of each group of...

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials or algebraic expressions. Understanding these is crucial for operations like addition, subtraction, and simplification, which often require common denominators.

Denominators in Algebraic Fractions

The denominator of an algebraic fraction holds the key to determining restrictions on the variable and plays a central role when combining or comparing expressions. Recognizing the factors in denominators helps in identifying the appropriate common denominator.

Least Common Denominator (LCD)

The least common denominator is the smallest expression that contains all the factors of each denominator, utilizing the highest powers of each distinct factor found in any denominator. This concept is essential for adding, subtracting, or comparing rational expressions.

Exponent Rules

Exponent rules are applied when analyzing the powers of variables within denominators. These rules help in determining the highest exponent for each factor, which is crucial in forming the LCD when working with algebraic expressions.

Equivalent Rational Expressions

Creating equivalent rational expressions involves rewriting each fraction so that they share the same denominator, typically the LCD. This is done by multiplying the numerator and denominator of each fraction by the necessary factors to achieve the common denominator without changing the value of the expression.

Transcript

00:01 So we're given two functions or two rational expressions, and we're asked to find the least common denominator, as well as create new expressions with that least common denominator that are equal to the original ones.

00:13 So least common denominator.

00:15 So to do that, let's factor out both denominators.

00:18 We have 6a squared b to the fourth, and this becomes 3 times 2 a squared p to the fourth, which becomes 2 times a squared, a squared, times 2.

00:31 B to the fourth and a squared times b to the fourth.

00:37 So we don't have to factor those out because, you know, the unique factor of a occurs twice because it's raised the power of two, and b occurs four times because it's raised the power of four.

00:51 And the other two unique factors are three and two, both of which occur once.

00:59 And now looking at the second denominator, we have a to the fourth, b which you can factor out as a to the 4 times b and so we see the only two factors are a and b here and a occurs four times because it's raised the power four and b occurs once so now you can find the least common denominator by looking at all the unique factors between these two denominators which are a b three and two so the most number greatest number of times a occurs between both denominators is four and for b it is two it is four it is four as well on the first diameter.

01:36 So let's raise it to the power four.

01:38 And for three and two, it occurs the greatest number times in the first denominator with once for both.

01:46 So we multiply all of these together.

01:47 We get six, a to the fourth, b to the fourth.

01:52 And now that we have our least common denominator, we can now try to turn these denominators into new rational expressions.

02:02 So now we have five by device...

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