$\left(4 M+N^2\right)^3=$

We are asked to expand the expression ( 4 M + N 2 ) 3 .
Apply the binomial expansion formula: ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 , where a = 4 M and b = N 2 .
Simplify each term in the expansion.
Combine the simplified terms to obtain the final expanded expression: 64 M 3 + 48 M 2 N 2 + 12 M N 4 + N 6 ​ .

Explanation

Understanding the Problem We are asked to expand the expression ( 4 M + N 2 ) 3 . This is a binomial expansion, and we can use the binomial theorem or the formula for the cube of a binomial to expand it. The formula is ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 . In our case, a = 4 M and b = N 2 .

Applying the Binomial Expansion Formula Now, let's substitute a = 4 M and b = N 2 into the formula ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 :
( 4 M + N 2 ) 3 = ( 4 M ) 3 + 3 ( 4 M ) 2 ( N 2 ) + 3 ( 4 M ) ( N 2 ) 2 + ( N 2 ) 3

Simplifying Each Term Next, we simplify each term:



( 4 M ) 3 = 4 3 × M 3 = 64 M 3
3 ( 4 M ) 2 ( N 2 ) = 3 ( 16 M 2 ) ( N 2 ) = 48 M 2 N 2
3 ( 4 M ) ( N 2 ) 2 = 3 ( 4 M ) ( N 4 ) = 12 M N 4
( N 2 ) 3 = N 2 × 3 = N 6


Combining the Terms Finally, we combine the simplified terms to get the final expanded expression: ( 4 M + N 2 ) 3 = 64 M 3 + 48 M 2 N 2 + 12 M N 4 + N 6

Examples
Expanding binomial expressions like ( 4 M + N 2 ) 3 is useful in various fields, such as physics and engineering, where polynomial expansions are used to approximate complex functions or model physical systems. For example, in fluid dynamics, the velocity potential of a fluid flow can be expressed as a polynomial, and expanding such polynomials helps in analyzing the flow behavior. Similarly, in structural mechanics, stress and strain relationships can be modeled using polynomial expansions, aiding in the design and analysis of structures. Understanding binomial expansion provides a foundation for these advanced applications.

Answered by GinnyAnswer | 2025-07-08