Expand and combine like terms.
$(8 x^4+1)^2=$
$\square$
Answer (1)
Recognize the expression as a binomial squared.
Apply the formula ( a + b ) 2 = a 2 + 2 ab + b 2 .
Simplify each term in the expanded expression.
Combine like terms to obtain the final answer: 64 x 8 + 16 x 4 + 1 .
Explanation
Understanding the Problem We are asked to expand the expression ( 8 x 4 + 1 ) 2 and combine like terms. This expression is a binomial squared, so we can use the formula ( a + b ) 2 = a 2 + 2 ab + b 2 .
Applying the Formula In this case, a = 8 x 4 and b = 1 . Substituting these values into the formula, we get ( 8 x 4 + 1 ) 2 = ( 8 x 4 ) 2 + 2 ( 8 x 4 ) ( 1 ) + ( 1 ) 2 .
Simplifying the Terms Now, we simplify each term:
( 8 x 4 ) 2 = 8 2 ⋅ ( x 4 ) 2 = 64 x 8
2 ( 8 x 4 ) ( 1 ) = 16 x 4
( 1 ) 2 = 1
So, we have 64 x 8 + 16 x 4 + 1 .
Combining Like Terms Finally, we combine the terms to get the expanded expression: 64 x 8 + 16 x 4 + 1 . There are no like terms to combine, so this is our final answer.
Final Answer Therefore, the expanded form of ( 8 x 4 + 1 ) 2 is 64 x 8 + 16 x 4 + 1 .
Examples
Understanding how to expand binomials like ( 8 x 4 + 1 ) 2 is useful in many areas of math and science. For example, in physics, you might encounter similar expressions when calculating the energy levels of a quantum system. In computer graphics, polynomial expressions are used to define curves and surfaces. Knowing how to quickly expand and simplify these expressions can save time and improve accuracy in these fields. This skill is also fundamental for more advanced algebraic manipulations and calculus problems.
