Question & Answer
QUESTION

Find the value of $8{{x}^{3}}+27{{y}^{3}}$ if $2x+3y=13$ and $xy=6$.

ANSWER Verified Verified
Hint: To find the value of the given algebraic expression, use the algebraic identity ${{\left( a+b \right)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right)$. Substitute $a=2x,b=3y$ in the algebraic identity and simplify the equation using the equations given in the question to get the value of $8{{x}^{3}}+27{{y}^{3}}$.

Complete step-by-step answer:
We know that for two variables ‘x’ and ‘y’, we have $2x+3y=13$ and $xy=6$. We have to calculate the value of the algebraic expression $8{{x}^{3}}+27{{y}^{3}}$.
To do so, we will use the algebraic identity ${{\left( a+b \right)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right)$.
Substituting $a=2x,b=3y$ in the above algebraic identity, we have ${{\left( 2x+3y \right)}^{3}}={{\left( 2x \right)}^{3}}+{{\left( 3y \right)}^{3}}+3\left( 2x \right)\left( 3y \right)+\left( 2x+3y \right)$.
Simplifying the above equation, we have ${{\left( 2x+3y \right)}^{3}}={{\left( 2x \right)}^{3}}+{{\left( 3y \right)}^{3}}+3\left( 2x \right)\left( 3y \right)+\left( 2x+3y \right)=8{{x}^{3}}+27{{y}^{3}}+18xy\left( 2x+3y \right)$.
Substituting $2x+3y=13$ and $xy=6$ in the above equation ${{\left( 13 \right)}^{3}}=8{{x}^{3}}+27{{y}^{3}}+18\times 6\times 13$.
Simplifying the above equation, we have $17069=8{{x}^{3}}+27{{y}^{3}}+1404$.
Rearranging the terms of the above equation, we have $8{{x}^{3}}+27{{y}^{3}}=17069-1404=15665$.
Hence, the value of the expression $8{{x}^{3}}+27{{y}^{3}}$ is 15665.
An algebraic identity is an equality that holds for all possible values of its variables. We can prove each of the identities by performing basic algebraic operations such as addition, multiplication, subtraction, and division. They are used for the factorization of the polynomials. That’s why they are useful in the computation of algebraic expressions. An algebraic expression differs from an algebraic identity in the way that the value of algebraic expression changes with the change in variables. However, an algebraic identity is equality which holds for all possible values of variables.

Note: We can also calculate the value of given algebraic expression by solving the linear equations $2x+3y=13$ and $xy=6$ by elimination method and finding the values of variables ‘x’ and ‘y’. then substitute those values in the given algebraic expression and calculate its value. Another way to solve this question is by using the algebraic identity ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}+{{b}^{2}}-ab \right)$.