\frac{3 y-2}{8 y} \div \frac{9 y^2-4}{6 y+4}
Answer (2)
Rewrite the division as multiplication by the reciprocal: 8 y 3 y − 2 × 9 y 2 − 4 6 y + 4 .
Factor the numerator and denominator: 8 y 3 y − 2 × ( 3 y − 2 ) ( 3 y + 2 ) 2 ( 3 y + 2 ) .
Cancel common factors: 8 y 3 y − 2 × ( 3 y − 2 ) ( 3 y + 2 ) 2 ( 3 y + 2 ) = 8 y 2 .
Simplify the expression: 8 y 2 = 4 y 1 . The final answer is 4 y 1 .
Explanation
Understanding the Problem We are asked to simplify the expression 8 y 3 y − 2 ÷ 6 y + 4 9 y 2 − 4 . This involves dividing one rational expression by another. To do this, we will rewrite the division as multiplication by the reciprocal. Then we will factor the numerators and denominators of the rational expressions and cancel any common factors.
Rewriting and Factoring First, rewrite the division as multiplication by the reciprocal: 8 y 3 y − 2 ÷ 6 y + 4 9 y 2 − 4 = 8 y 3 y − 2 × 9 y 2 − 4 6 y + 4 Next, factor the numerator of the second fraction: 6 y + 4 = 2 ( 3 y + 2 ) Then, factor the denominator of the second fraction, which is a difference of squares: 9 y 2 − 4 = ( 3 y − 2 ) ( 3 y + 2 ) Now, rewrite the expression with the factored terms: 8 y 3 y − 2 × ( 3 y − 2 ) ( 3 y + 2 ) 2 ( 3 y + 2 )
Canceling and Simplifying Now, cancel common factors in the numerator and denominator. We can cancel the factor ( 3 y − 2 ) and the factor ( 3 y + 2 ) :
8 y 3 y − 2 × ( 3 y − 2 ) ( 3 y + 2 ) 2 ( 3 y + 2 ) = 8 y 1 × 2 Simplify the resulting expression: 8 y 1 × 2 = 8 y 2 = 4 y 1
Final Answer Therefore, the simplified expression is 4 y 1 .
Examples
Rational expressions are useful in many areas of science and engineering. For example, in electrical engineering, the impedance of a circuit can be represented by a rational expression. Simplifying such expressions can help engineers analyze and design circuits more efficiently. Also, in physics, rational expressions can appear when dealing with lenses and optics, where simplifying them helps in calculating image distances and magnifications.
The expression 8 y 3 y − 2 ÷ 6 y + 4 9 y 2 − 4 simplifies to 4 y 1 . This involves rewriting the division as multiplication by the reciprocal, factoring, canceling common terms, and simplifying. The final answer is 4 y 1 .
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