(h) [tex]\frac{a^4}{4(a+3)^2}+\frac{81-18 a^2}{4(a+3)^2}[/tex]
Answer (1)
Combine the fractions since they have the same denominator: 4 ( a + 3 ) 2 a 4 + 81 − 18 a 2 .
Factor the numerator: 4 ( a + 3 ) 2 ( a 2 − 9 ) 2 .
Further factor the numerator: 4 ( a + 3 ) 2 ( a − 3 ) 2 ( a + 3 ) 2 .
Cancel out the common factor: 4 ( a − 3 ) 2 .
The simplified expression is: 4 ( a − 3 ) 2 .
Explanation
Understanding the Problem We are given the expression 4 ( a + 3 ) 2 a 4 + 4 ( a + 3 ) 2 81 − 18 a 2 Our goal is to simplify this expression.
Combining Fractions Since the two fractions have the same denominator, we can combine them into a single fraction: 4 ( a + 3 ) 2 a 4 + 81 − 18 a 2 Rearranging the terms in the numerator, we get: 4 ( a + 3 ) 2 a 4 − 18 a 2 + 81
Factoring the Numerator Notice that the numerator is a perfect square trinomial. We can rewrite it as: a 4 − 18 a 2 + 81 = ( a 2 − 9 ) 2 So the expression becomes: 4 ( a + 3 ) 2 ( a 2 − 9 ) 2
Further Factoring We can further factor the term a 2 − 9 as a difference of squares: a 2 − 9 = ( a − 3 ) ( a + 3 ) So, ( a 2 − 9 ) 2 = [( a − 3 ) ( a + 3 ) ] 2 = ( a − 3 ) 2 ( a + 3 ) 2 . The expression now looks like this: 4 ( a + 3 ) 2 ( a − 3 ) 2 ( a + 3 ) 2
Canceling Common Factors We can cancel out the common factor of ( a + 3 ) 2 from the numerator and the denominator, assuming a = − 3 :
4 ( a − 3 ) 2
Expanding and Simplifying Expanding the numerator, we get: 4 a 2 − 6 a + 9 = 4 a 2 − 4 6 a + 4 9 = 4 a 2 − 2 3 a + 4 9 This can also be written as: 4 1 ( a − 3 ) 2
Final Answer Therefore, the simplified expression is: 4 ( a − 3 ) 2 = 4 a 2 − 2 3 a + 4 9
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics with applications in various fields. For instance, in physics, you might encounter complex equations describing the motion of objects. Simplifying these equations can make them easier to analyze and solve. Similarly, in engineering, simplifying expressions can help optimize designs and reduce computational complexity. For example, consider a scenario where you are designing a bridge and need to calculate the stress on a particular beam. The equation for stress might be initially complex, but by simplifying it, you can more easily determine the optimal dimensions of the beam to ensure it can withstand the load.
