Find the product.
$(3 a^4+4)^2$
A. $6 a^4+8$
B. $9 a^8+16$
C. $9 a^8+24 a^4+16$
D. $9 a^{16}+24 a^4+16$
Answer (2)
We need to find the product of ( 3 a 4 + 4 ) 2 .
Apply the formula ( x + y ) 2 = x 2 + 2 x y + y 2 with x = 3 a 4 and y = 4 .
Simplify the expression: ( 3 a 4 ) 2 + 2 ( 3 a 4 ) ( 4 ) + ( 4 ) 2 = 9 a 8 + 24 a 4 + 16 .
The final product is 9 a 8 + 24 a 4 + 16 .
Explanation
Understanding the Problem We are asked to find the product of the expression ( 3 a 4 + 4 ) 2 . This means we need to expand the expression and simplify it.
Applying the Square of a Binomial Formula To find the product, we will use the formula for the square of a binomial: ( x + y ) 2 = x 2 + 2 x y + y 2 . In our case, x = 3 a 4 and y = 4 .
Substituting Values Now, we substitute x and y into the formula: ( 3 a 4 + 4 ) 2 = ( 3 a 4 ) 2 + 2 ( 3 a 4 ) ( 4 ) + ( 4 ) 2
Simplifying the Expression Next, we simplify each term:
( 3 a 4 ) 2 = 9 a 8
2 ( 3 a 4 ) ( 4 ) = 24 a 4
( 4 ) 2 = 16
So, the expanded expression is: 9 a 8 + 24 a 4 + 16
Final Answer Therefore, the product of ( 3 a 4 + 4 ) 2 is 9 a 8 + 24 a 4 + 16 .
Examples
Understanding how to expand and simplify expressions like ( 3 a 4 + 4 ) 2 is crucial in various fields, such as physics and engineering, where complex equations often need to be manipulated. For instance, when calculating the energy of a system or designing a structure, you might encounter similar expressions. Being able to quickly and accurately expand and simplify these expressions can save time and reduce errors in your calculations. This skill also forms the basis for more advanced algebraic manipulations and problem-solving techniques.
To find the product of ( 3 a 4 + 4 ) 2 , we apply the square of a binomial formula which gives us 9 a 8 + 24 a 4 + 16 . The correct answer is option C. We expand and simplify each term in the expression to arrive at this result.
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