Multiplying/Dividing Rational Expressions

Question

Simplify:

(2x^9/15y^4)/(22x^8/35y^2)=

Answer 1

Background

First, I would say to remember that dividing by a fraction is equivalent to multiplying by its reciprocal.

For example: (a/b)/(c/d) = (a/b)*(d/c) = ad/bc

Additionally, when dividing exponents, you subtract the powers.

For example,

x^a/x^b = x^(a-b)

Conversely, when multiplying exponents, you add the powers.

For example,

x^a * x^b = x^(a+b)

Solving the question

Applying this to the question at hand,

(2x^9/15y^4)/(22x^8/35y^2) = (2x^9/15y^4)*(35y^2/22x^8)

= (2 * x^9 * 35 * y^2)/(15 * y^4 * 22 * x^8)

= (2 * x^9 * 35 * y^2)/(15 * y^4 * 22 * x^8)

= (70 * x^9 * y^2)(330x^8y^4)

= (7 * x^9 * y^2)/(33 * x^8 * y^4)

= (7 * x)/(33 * y^2)

Solution

(2x^9/15y^4)/(22x^8/35y^2) = 7x/33y^2

Answer 2

Starting with (2x⁹/15y⁴)/(22x⁸/35y²)

Dividing by a fraction is the same as multiplying by its reciprocal

(a/b)/(c/d) = a/b × d/c = ad/bc

Applying this to the rational expression we are trying to simplify:

(2x⁹/15y⁴)/(22x⁸/35y²) = (2x⁹)/(15y⁴) × (35y²)/(22x⁸) = (2x⁹35y²)/(15y⁴22x⁸) = (70x⁹y²)/(330x⁸y⁴)

Rewrite as the product of three fractions, each with like terms in the numerator and denominator.

(70x⁹y²)/(330x⁸y⁴) = 70/330 × x⁹/x⁸ × y²/y⁴ = (7/33)(x^9–8)(y^2–4) = (7/33)(x)(y^–2)

(Noting that x^a/x^b = x^a–b)

Noting that y^–2 = 1/y², make this substitution

(7/33)(x)(y^–2) = (7/33)(x)(1/y²)

Now multiply to express as a single fraction

(7/33)(x)(1/y²) = (7x)/(33y²)

So simplifying (2x⁹/15y⁴)/(22x⁸/35y²) gives

(7x)/(33y²)