Multiply using the Binomial Squares Pattern.
$\left(5 a^3-1\right)^2=$ $\square$
Answer (1)
Recognize the binomial squares pattern: ( x − y ) 2 = x 2 − 2 x y + y 2 .
Substitute x = 5 a 3 and y = 1 into the pattern.
Simplify the expression: ( 5 a 3 ) 2 − 2 ( 5 a 3 ) ( 1 ) + ( 1 ) 2 = 25 a 6 − 10 a 3 + 1 .
The final answer is 25 a 6 − 10 a 3 + 1 .
Explanation
Understanding the Problem We are asked to multiply ( 5 a 3 − 1 ) 2 using the binomial squares pattern. The binomial squares pattern states that for any two terms x and y , ( x − y ) 2 = x 2 − 2 x y + y 2 . In our case, x = 5 a 3 and y = 1 . We will substitute these values into the binomial squares pattern and simplify.
Applying the Binomial Squares Pattern Now, let's substitute x = 5 a 3 and y = 1 into the binomial squares pattern: ( 5 a 3 − 1 ) 2 = ( 5 a 3 ) 2 − 2 ( 5 a 3 ) ( 1 ) + ( 1 ) 2
Simplifying the Expression Next, we simplify each term:
( 5 a 3 ) 2 = 5 2 ⋅ ( a 3 ) 2 = 25 a 6
− 2 ( 5 a 3 ) ( 1 ) = − 10 a 3
( 1 ) 2 = 1
So, the expression becomes: 25 a 6 − 10 a 3 + 1
Final Answer Therefore, ( 5 a 3 − 1 ) 2 = 25 a 6 − 10 a 3 + 1 .
Examples
The binomial squares pattern is useful in many areas of mathematics and physics. For example, when calculating areas or volumes, you might encounter expressions that can be simplified using this pattern. Imagine you are designing a square garden with side length ( x − 2 ) . The area of the garden would be ( x − 2 ) 2 . Using the binomial squares pattern, you can expand this to x 2 − 4 x + 4 , which helps you understand how the area changes as x changes. This pattern is also essential in algebra for simplifying and solving equations.
