\frac{3 a b^2}{5 x y} \div \frac{12 a b-6 a}{x^2 y+2 x y^2}
Answer (2)
To simplify the given expression, rewrite the division as multiplication of the reciprocal. Factor the numerator and denominator where possible, cancel common factors, and simplify the constants. The final expression simplifies to 10 ( 2 b − 1 ) b 2 ( x + 2 y ) .
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Rewrite the division as multiplication by the reciprocal: 5 x y 3 a b 2 ÷ x 2 y + 2 x y 2 12 ab − 6 a = 5 x y 3 a b 2 ⋅ 12 ab − 6 a x 2 y + 2 x y 2 .
Factor the numerator and denominator of the second fraction: x 2 y + 2 x y 2 = x y ( x + 2 y ) and 12 ab − 6 a = 6 a ( 2 b − 1 ) .
Substitute the factored expressions into the expression: 5 x y 3 a b 2 ⋅ 6 a ( 2 b − 1 ) x y ( x + 2 y ) .
Cancel common factors and simplify: 10 ( 2 b − 1 ) b 2 ( x + 2 y ) .
The simplified expression is 10 ( 2 b − 1 ) b 2 ( x + 2 y ) .
Explanation
Understanding the Problem We are asked to simplify the expression 5 x y 3 a b 2 ÷ x 2 y + 2 x y 2 12 ab − 6 a . This involves dividing one rational expression by another, which is equivalent to multiplying by the reciprocal of the second expression. We will factor and cancel common terms to simplify.
Rewriting as Multiplication First, we rewrite the division as multiplication by the reciprocal: 5 x y 3 a b 2 ÷ x 2 y + 2 x y 2 12 ab − 6 a = 5 x y 3 a b 2 ⋅ 12 ab − 6 a x 2 y + 2 x y 2
Factoring the Numerator Next, we factor the numerator of the second fraction: x 2 y + 2 x y 2 = x y ( x + 2 y )
Factoring the Denominator Then, we factor the denominator of the second fraction: 12 ab − 6 a = 6 a ( 2 b − 1 )
Substituting Factored Expressions Now, substitute the factored expressions back into the expression: 5 x y 3 a b 2 ⋅ 6 a ( 2 b − 1 ) x y ( x + 2 y )
Canceling Common Factors Next, we cancel common factors. We can cancel a , x , and y :
5 x y 3 a b 2 ⋅ 6 a ( 2 b − 1 ) x y ( x + 2 y ) = 5 3 b 2 ⋅ 6 ( 2 b − 1 ) ( x + 2 y )
Simplifying Constants Finally, we simplify the constants: 5 3 b 2 ⋅ 6 ( 2 b − 1 ) ( x + 2 y ) = 10 ( 2 b − 1 ) b 2 ( x + 2 y )
Final Answer Therefore, the simplified expression is 10 ( 2 b − 1 ) b 2 ( x + 2 y ) .
Examples
Simplifying rational expressions is useful in many areas of math and science. For example, in physics, you might use it to simplify an expression for the force between two objects. In calculus, you might use it to simplify an expression before integrating or differentiating. In everyday life, you might use it to simplify a ratio of two quantities to make it easier to understand.
