Expand and combine like terms.
$\left(4+7 a^2\right)^2=$
$\square$
Answer (1)
Identify the binomial expression: ( 4 + 7 a 2 ) 2 .
Apply the binomial expansion formula: ( x + y ) 2 = x 2 + 2 x y + y 2 .
Calculate each term: x 2 = 16 , 2 x y = 56 a 2 , y 2 = 49 a 4 .
Combine and write in standard form: 49 a 4 + 56 a 2 + 16 .
Explanation
Understanding the Problem We are asked to expand the expression ( 4 + 7 a 2 ) 2 and combine like terms. This expression is a binomial squared, so we can use the formula ( x + y ) 2 = x 2 + 2 x y + y 2 .
Identifying x and y Let's identify x and y in our expression. Here, x = 4 and y = 7 a 2 . Now we can apply the binomial expansion formula.
Calculating x^2 First, we calculate x 2 = 4 2 = 16 .
Calculating 2xy Next, we calculate 2 x y = 2 ( 4 ) ( 7 a 2 ) = 56 a 2 .
Calculating y^2 Then, we calculate y 2 = ( 7 a 2 ) 2 = 49 a 4 .
Combining the terms Now, we combine the terms: x 2 + 2 x y + y 2 = 16 + 56 a 2 + 49 a 4 .
Writing in Standard Form Finally, we write the expression in standard form, arranging the terms in descending order of the exponent of a : 49 a 4 + 56 a 2 + 16 .
Final Answer The expanded and simplified expression is 49 a 4 + 56 a 2 + 16 .
Examples
Understanding how to expand and simplify expressions like this is fundamental in algebra and calculus. For example, when calculating areas or volumes in physics or engineering, you often encounter squared expressions. Imagine you're designing a square garden with sides of length ( 4 + 7 a 2 ) . Expanding ( 4 + 7 a 2 ) 2 tells you the total area of the garden, which helps in planning the layout and resource allocation, such as fencing and soil.
