Question

How do you simplify \[3\dfrac{{{x}^{2}}}{4}\left( xy-z \right)+4x\dfrac{y}{3}\left( {{x}^{2}}+2 \right)-2\dfrac{{{x}^{2}}}{3}\left( xy-z \right)+3x\dfrac{y}{5}\left( {{x}^{2}}+2 \right)\]?

Answer 1

Hint: In this problem, we have to simplify the given expression. We can see that we have similar terms in the expression. We can first convert the mixed fraction into its improper fraction, we can then arrange the terms as there are similar terms in the expression. We can then multiply the terms inside the brackets and simplify the terms to get a simplified form.

Complete step by step solution:
We know that the given expression to be simplified is,
\[3\dfrac{{{x}^{2}}}{4}\left( xy-z \right)+4x\dfrac{y}{3}\left( {{x}^{2}}+2 \right)-2\dfrac{{{x}^{2}}}{3}\left( xy-z \right)+3x\dfrac{y}{5}\left( {{x}^{2}}+2 \right)\]
We can see that there are mixed fraction, we can now convert them into improper fraction, we get
\[\Rightarrow \dfrac{13{{x}^{2}}}{4}\left( xy-z \right)+\dfrac{13xy}{3}\left( {{x}^{2}}+2 \right)-\dfrac{7{{x}^{2}}}{3}\left( xy-z \right)+\dfrac{16xy}{5}\left( {{x}^{2}}+2 \right)\]
We can now multiply the terms inside the brackets we get,
\[\Rightarrow \dfrac{13{{x}^{3}}y}{4}-\dfrac{13{{x}^{2}}z}{4}+\dfrac{13{{x}^{3}}y}{3}+\dfrac{26xy}{3}-\dfrac{7{{x}^{3}}y}{3}+\dfrac{7{{x}^{2}}z}{3}+\dfrac{16{{x}^{3}}y}{5}+\dfrac{32xy}{5}\]
We can now arrange the terms in order which has same variables, we get
\[\Rightarrow -\dfrac{13{{x}^{2}}z}{4}+\dfrac{7{{x}^{2}}z}{3}+\dfrac{13{{x}^{3}}y}{3}-\dfrac{7{{x}^{3}}y}{3}+\dfrac{13{{x}^{3}}y}{4}+\dfrac{16{{x}^{3}}y}{5}+\dfrac{32xy}{5}+\dfrac{26xy}{3}\]
We can now simplify the terms, where we have similar numerator, but different numerator, so we can cross multiply the terms, to get a simplified form we get,
\[\Rightarrow \dfrac{-39{{x}^{2}}z+28{{x}^{2}}z}{12}+\dfrac{39{{x}^{3}}y-21{{x}^{3}}y}{9}+\dfrac{65{{x}^{3}}y+64{{x}^{3}}y}{20}+\dfrac{96xy+130xy}{15}\]
We can now simplify the above step by adding or subtracting the terms in the numerator, we get
\[\Rightarrow -\dfrac{11{{x}^{2}}z}{12}+\dfrac{507{{x}^{3}}y}{60}+\dfrac{226xy}{15}\]
Therefore, the simplified form of the given expression \[3\dfrac{{{x}^{2}}}{4}\left( xy-z \right)+4x\dfrac{y}{3}\left( {{x}^{2}}+2 \right)-2\dfrac{{{x}^{2}}}{3}\left( xy-z \right)+3x\dfrac{y}{5}\left( {{x}^{2}}+2 \right)\] is \[-\dfrac{11{{x}^{2}}z}{12}+\dfrac{507{{x}^{3}}y}{60}+\dfrac{226xy}{15}\].

Note: Students make mistakes while cross multiplying the terms we can cross multiply each term of the numerator and the denominator and write it in the numerator, we can then multiply both the denominator and write the result in the denominator. We should concentrate while multiplying the terms inside the bracket, we should also check for the correct sign is written or not.