Question

: Find the cube of $(2x + 3y)$

Answer 1

Hint: Here we will first observe the given expression and will convert the given expression in the form of the whole cube. Use the identity of the algebraic expression${(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)$ and then will simplify for the required answer.

Complete step-by-step answer:
Given expression: cube of $(2x + 3y)$
The mathematical expression can be written as: ${(2x + 3y)^3}$
Now, comparing and applying the formula for the binomial whole cube formula ${(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)$
${(2x + 3y)^3} = {(2x)^3} + {(3y)^3} + 3(2x)(3y)(2x + 3y)$
Cube of the term is expressed as the product of the same number twice. For example, the simplified form for the first term simplified form of cubed number is ${(2x)^3} = 2x \times 2x \times 2x = 8{x^3}$. Similarly apply the same concept for the second term.
${(2x + 3y)^3} = 8{x^3} + 27{y^3} + 18xy(2x + 3y)$
Term outside the bracket when brackets are opened is multiplied with all the terms inside the bracket.
${(2x + 3y)^3} = 8{x^3} + 27{y^3} + 18xy(2x) + 18xy(3y)$
Simplify the above equation finding the product of the terms.
${(2x + 3y)^3} = 8{x^3} + 27{y^3} + 36{x^2}y + 54x{y^2}$
This is the required solution.
So, the correct answer is “$8{x^3} + 27{y^3} + 36{x^2}y + 54x{y^2}$”.

Note: To find the expanded form of the algebraic expressions remember the identities relating to the sum and difference of cubes and sum and difference of squares formulas for an accurate and an efficient solution. Remember squares and cubes of numbers and remember them at least till twenty.