Question

Find the product of the given monomials:
\[xyz\]
\[{{x}^{2}}yz\]

Answer 1

Hint: To solve the given question, we will first find out what a monomial is and after that, we will assume that the value of xyz is ‘a’ and the value of \[{{x}^{2}}yz\] is b. Then we will multiply ‘a’ and ‘b’. On the right-hand side, we will write the same variables together. Then we will convert the two same variables to the power of that variable by using the exponential identity.

Complete step-by-step answer:
\[{{p}^{\alpha }}.{{p}^{\beta }}={{p}^{\alpha \beta }}\]

Before we solve the question, we will first find out what a monomial is. A monomial is a type of polynomial which has only one term. In other words, we can say that a monomial is a product of powers of the variables with non-negative integer exponents. In our question, the polynomials given are xyz and \[{{x}^{2}}yz.\] Let us assume that the value of xyz is ‘a’ and \[{{x}^{2}}yz\] is ‘b’. Thus, we have,
\[a=xyz\]
\[b={{x}^{2}}yz\]
Now, we will multiply ‘a’ and ‘b’. Thus, we will get,
\[\Rightarrow a\times b=\left( xyz \right)\times \left( {{x}^{2}}yz \right).....\left( i \right)\]
\[\Rightarrow a\times b=\left( xyz \right)\left( {{x}^{2}}yz \right)\]
Now, we will write similar terms together. Thus, we will get,
\[\Rightarrow a\times b=\left( x\times {{x}^{2}} \right)\left( y\times y \right)\left( z\times z \right)\]
Now, we will use the following exponential identity:
\[{{p}^{\alpha }}.{{p}^{\beta }}={{p}^{\alpha \beta }}\]
Thus, we will get,
\[\Rightarrow a\times b=\left( {{x}^{1+2}} \right)\left( {{y}^{1+1}} \right)\left( {{z}^{1+1}} \right)\]
\[\Rightarrow a\times b=\left( {{x}^{3}} \right)\left( {{y}^{2}} \right)\left( {{z}^{2}} \right)\]
\[\Rightarrow a\times b={{x}^{3}}{{y}^{2}}{{z}^{2}}.....\left( ii \right)\]
From (i) and (ii), we will get the following equation:
\[\Rightarrow \left( xyz \right)\times \left( {{x}^{2}}yz \right)={{x}^{3}}{{y}^{2}}{{z}^{2}}\]
Hence, the product of the monomials xyz and \[{{x}^{2}}yz\] is \[{{x}^{3}}{{y}^{2}}{{z}^{2}}.\]

Note: We cannot multiply the monomials like we multiply polynomials. For example, if there are two polynomials (a + b) and (c + d), then their product will be \[\left( a+b \right)\left( c+d \right)=a\left( c+d \right)+b\left( c+d \right).\] If we have two monomials, ab and cd, then we cannot write their product as shown below.
\[\left( ab \right)\left( cd \right)=a\left( cd \right)\times b\left( cd \right)=ab{{c}^{2}}{{d}^{2}}\]
This is the wrong method.