Question

How do you simplify ${\left( {\dfrac{{3x{y^4}}}{{5{z^2}}}} \right)^2}$?

Answer 1

Hint: We know that while calculating the product of two algebraic expressions we have to multiply the constants separately and variables separately. This will give us a product of two algebraic expressions. Let us assume the first algebraic expression is $3{x^2}y$ and the second algebraic expression is $4x{y^3}$. Let us assume the constant part of $3{x^2}y$ is equal to ${C_1}$. Similarly, let us assume the constant part of $4x{y^3}$ is equal to ${C_2}$. Now we have to find the product of ${C_1}$ and ${C_2}$. Let us assume this product as ${C_3}$. Let us assume the variable part of $3{x^2}y$ is equal to ${V_1}$. Similarly, let us assume the constant part of $4x{y^3}$ is equal to ${V_2}$. Now we have to find the product of ${V_1}$ and ${V_2}$. Let us assume this product as ${V_3}$. Let us assume the product of given algebraic expressions is equal to C. Now we have to find the product of ${C_3}$ and ${V_3}$. This gives us the algebraic expression C.

Complete step-by-step solution:
Before solving the question, we should know that while calculating the product of two algebraic expressions we have to multiply the constants separately and variables separately. This will give us a product of two algebraic expressions.
From the question, we were given that to find the square of $\dfrac{{3x{y^4}}}{{5{z^2}}}$ means the product of $\dfrac{{3x{y^4}}}{{5{z^2}}}$ with itself. Now we have to divide the given algebraic expressions into two parts. Let us consider the algebraic expression as A. From the given question, it was given that the first algebraic expression is $\dfrac{{3x{y^4}}}{{5{z^2}}}$.
First algebraic expression:
$ \Rightarrow A = \dfrac{{3x{y^4}}}{{5{z^2}}}$................….. (1)
Now we have to divide the equation into two parts where the first part represents a constant and the second part indicates variables.
In equation (1), $\dfrac{3}{5}$ is the constant part and $\dfrac{{x{y^4}}}{{{z^2}}}$ is the variable part.
Let us assume the constant part is equal to ${C_1}$
$ \Rightarrow {C_1} = \dfrac{3}{5}$..............….. (2)
Let us assume the variable part is equal to ${V_1}$
$ \Rightarrow {V_1} = \dfrac{{x{y^4}}}{{{z^2}}}$...............….. (3)
Now to find the product of the algebraic expression with itself, we have to find the product of the constant parts of both algebraic expressions and the product of the variable parts of both algebraic expressions.
Let us assume the product of the constant part is equal to ${C_3}$.
Now we have to find the square of ${C_1}$.
$ \Rightarrow {C_3} = {C_1}^2$
Now we will substitute value from equation (2) in the above equation, we get
$ \Rightarrow {C_3} = {\left( {\dfrac{3}{5}} \right)^2}$
Square the terms,
$ \Rightarrow {C_3} = \dfrac{9}{{25}}$.............….. (4)
Let us assume the product of the variable part is equal to ${V_3}$.
Now we have to find the square of ${V_1}$.
$ \Rightarrow {V_3} = {V_1}^2$
Now we will substitute the value from equation (3) in the above equation, we get
$ \Rightarrow {V_3} = {\left( {\dfrac{{x{y^4}}}{{{z^2}}}} \right)^2}$
Square the terms,
$ \Rightarrow {V_3} = \dfrac{{{x^2}{y^8}}}{{{z^4}}}$............….. (5)
We know that the product of the constant part of an algebraic expression and the variable part of an algebraic expression gives us the required algebraic expression.
Similarly, the product of the constant part and the variable part gives us the algebraic expression.
$ \Rightarrow C = {C_3}{V_3}$
Now we have to substitute equation (4) and equation (5), we get
$ \Rightarrow C = \dfrac{9}{{25}} \times \dfrac{{{x^2}{y^8}}}{{{z^4}}}$
Simplify the terms.
$ \Rightarrow C = \dfrac{{9{x^2}{y^8}}}{{25{z^4}}}$

Hence, the value is $\dfrac{{9{x^2}{y^8}}}{{25{z^4}}}$.

Note: Algebraic expressions explain a set of operations that should be done following a specific set of orders. Such expressions consist of an amalgamation of integers, variables, exponents, and constants. When these expressions undergo the mathematical operation of multiplication, then the process is called the multiplication of algebraic expression. Two different expressions that give the same answer are called equivalent expressions. Some other properties like distributive and commutative property of addition will come in handy while doing multiplying polynomials.