Question

How do you simplify ${(2x + 3y)^3}$ ?

Answer 1

Hint: The given expression is an algebraic expression as it is a combination of numerical values and unknown variable quantities represented by an alphabet. In this question, x and y are used to express some unknown variable quantity. We don’t have to find the value of x and y, we have to find the cube of the sum of 2x and 3y. Cube of a number means that number multiplied with itself 3 times, for example, 2 multiplied with itself three times gives 8, so 8 is the cube of 2. Using the above-mentioned definitions, we can simplify the given algebraic expression.

Complete step by step answer:
We have to simplify ${(2x + 3y)^3}$ that is we have to find the cube of $2x + 3y$
We know that –
$
\Rightarrow {(a + b)^3} = {a^3} + {b^3} + 3ab(a + b) \\
   \Rightarrow {(2x + 3y)^3} = {(2x)^3} + {(3y)^3} + 3 \times 2x \times 3y(2x + 3y) \\
   \Rightarrow {(2x + 3y)^3} = 8{x^3} + 27{y^3} + 18xy(2x + 3y) \\
   \Rightarrow {(2x + 3y)^3} = 8{x^3} + 27{y^3} + 36{x^2}y + 54x{y^2} \\
 $
Hence, the simplified form of ${(2x + 3y)^3}$ is $8{x^3} + 27{y^3} + 36{x^2}y + 54x{y^2}$.

Note: The simplified form of an expression means a more understandable and easier way of writing the expression. We can find the cube of $2x + 3y$ by writing ${(2x + 3y)^3}$ as $(2x + 3y)(2x + 3y)(2x + 3y)$ then we can multiply the first two brackets with each other and then multiply the obtained equation with the third bracket but this process turns out to be quite long so there are several formulas for simplifying such expressions. We have used the formula that states, the cube of the sum of two numbers is equal to the cube of the first number plus the cube of the second number plus the product of 3, first number, second number and the sum of the two numbers, that is, ${(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)$.