In this post we will learn the process of multiplying monomials.
To understand the concept, you should have basic knowledge of constants, variables, powers and exponents.
Method for Multiplying Monomials
Before understanding the process, let us revise the basic concepts.
What are monomials?
It’s an entity which is made of constants, variables or both.
Example of monomials are:
\mathtt{\Longrightarrow 2}\\\ \\ \mathtt{\Longrightarrow \ 9x}\\\ \\ \mathtt{\Longrightarrow \ 15xy^{2}}\\\ \\ \mathtt{\Longrightarrow \ -17\ x^{3} yz^{2}}
Note that there is only one entity with no addition or subtraction signs.
Steps to Multiply two monomials
To multiply two or more monomials, follow the below rules:
(a) The coefficients should be multiplied together.
(b) Same variables are multiplied using the rule of exponents.
Important Exponent rules used in this chapter are:
⟹ Exponents of same base can multiplied by adding the powers.
⟹ Exponents raised to another power is simplified by multiplication of powers.
The above rules are sufficient to solve monomial multiplication problem.
Let us see some examples for our understanding.
Multiplying Monomial Examples
Examples 01
Multiply \mathtt{4y^{2} \ \&\ 5y}
Solution
\mathtt{\Longrightarrow 4y^{2} \ \times \ 5y}
Separate the coefficients and similar variables.
\mathtt{\Longrightarrow \ ( 4\ \times \ 5) \ \times \left( y^{2} \times y\right)}
Multiply the separated coefficients and variables
\mathtt{\Longrightarrow \ 20\ \times \ \left( y^{2+1}\right)}\\\ \\ \mathtt{\Longrightarrow \ 20\times \ y^{3}}\\\ \\ \mathtt{\Longrightarrow \ 20y^{3}}
Example 02
Multiply \mathtt{3x^{3} y^{2} \ \&\ \ 15x^{5} y^{6}}
Solution
\mathtt{\Longrightarrow 3x^{3} y^{2} \ \times \ \ 15x^{5} y^{6}}
Separate the coefficients and same variables
\mathtt{\Longrightarrow \ ( 3\ \times \ 15) .\left( x^{3} \times x^{5}\right) .\left( y^{2} \times y^{6}\right)}
Now multiply the coefficients and variables using above mentioned exponent rule
\mathtt{\Longrightarrow \ ( 45) \ .\left( x^{3+5}\right) .\left( y^{2+6}\right)}\\\ \\ \mathtt{\Longrightarrow \ 45.\ x^{8} .y^{8}}\\\ \\ \mathtt{\Longrightarrow \ 45x^{8} y^{8}}
Example 03
Multiply \mathtt{\left( 7x^{2} y^{3}\right)^{2} \ \ \&\ \ -4xyz{^{2}}}
Solution
Note that the first monomial is raised to second power.
Let us simplify this monomial before multiplying it with the second
\mathtt{\Longrightarrow \ \left( 7x^{2} y^{3}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 7^{2} .\ \left( x^{2}\right)^{2} \ .\ \left( y^{3}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 49\ .\ ( x)^{2\times 2} .\ y^{3\times 2}}\\\ \\ \mathtt{\Longrightarrow \ 49\ .\ x^{4} .\ y^{6}}\\\ \\ \mathtt{\Longrightarrow \ 49x^{4} y^{6}}
Now multiply \mathtt{49x^{4} y^{6} \ \ \ \&\ -4xyz^{2}}
Separating the coefficients and same variables
\mathtt{\Longrightarrow \ ( 49\times \ -4)\left( x^{4} \times x\right)\left( y^{6} \times y\right) .z^{2}}\\\ \\ \mathtt{\Longrightarrow \ -196.\ \left( x^{4\ +1}\right)\left( y^{6+1}\right) z^{2}}\\\ \\ \mathtt{\Longrightarrow \ -196\ x^{5} y^{7} z^{2}}
Example 4
Multiply \mathtt{9a^{2} b^{3} \ \&\ \ 3b^{6} c^{3} \ \ \&\ \ 2a^{4} bc^{5} \ }
Solution
\mathtt{\Longrightarrow 9a^{2} b^{3} \times \ 3b^{6} c^{3} \times 2a^{4} bc^{5} \ }
Separate the coefficient and same variables.
\mathtt{\Longrightarrow ( 9\times 3\times 2)\left( a^{2} \times a^{4}\right)\left( b^{3} \times b^{6} \times b\right)\left( c^{3} \times c^{5}\right)}\\\ \\ \mathtt{\Longrightarrow \ 54.\ \left( a^{2\ +4}\right)\left( b^{3+6+1}\right)\left( c^{3+5}\right)}\\\ \\ \mathtt{\Longrightarrow \ 54\ a^{6} b^{10} c^{8}}Example 05
Multiply \mathtt{\left( 10xy^{4}\right)^{2} \&\ \left(\frac{2}{3} x^{2} y\right)^{3}}
Solution
Both the monomials are raised to second power.
Let us simplify the monomials first.
\mathtt{\Longrightarrow \left( 10xy^{4}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 10^{2} x^{2} y^{4\times 2}}\\\ \\ \mathtt{\Longrightarrow \ 100\ x^{2} y^{8}}
Simplifying Second Monomial
\mathtt{\Longrightarrow \left(\frac{2}{3} x^{2} y\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2^{3}}{3^{3}} x^{2\times 3} \ y^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8}{27} \ x^{6} \ y^{3}}
Now Multiply \mathtt{100x^{2} y^{8\ \ } \&\ \ \frac{8}{27} \ x^{6} \ y^{3}}
Separate the coefficients and same variable
\mathtt{\Longrightarrow \ \left( 100\times \frac{8}{27}\right)\left( x^{2} \times x^{3}\right)\left( y^{8} \times y^{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{800}{27}\left( x^{2+3}\right)\left( y^{8+3}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{800}{27} x^{5} y^{11}}
Frequently asked Questions – Multiplying Monomials
(01) What’s the difference between monomial and binomial?
Monomial is an algebraic expression with one entity.
Example:
\mathtt{\Longrightarrow \ 3x}\\ \\ \mathtt{\Longrightarrow \ 10x^{2} y}\\ \\ \mathtt{\Longrightarrow \ 13}
Binomial is an algebraic expression with two entities.
The entities are separated by addition or subtraction sign.
Examples:
\mathtt{\Longrightarrow \ x\ +\ 3}\\ \\ \mathtt{\Longrightarrow \ 3x^{2} +\ 2y}\\ \\ \mathtt{\Longrightarrow \ x^{2} y\ +\ xy^{2}}
(02) Is number 4 a monomial?
Yes!!
Its a algebraic expression made of constant entity 4.
(03) The length of rectangle is 6xy and breadth is \mathtt{3xy^{2}} . Find the area of rectangle.
Its given that;
Length = 6xy
Breadth = \mathtt{3xy^{2}}
We know that;
Area of Rectangle = Length x Breadth
Area = \mathtt{6xy\ \times \ 3xy^{2}}
Here we have to multiply the two monomials.
\mathtt{Area\ =\ ( 6\times 3)( x\times x)\left( y\times y^{2}\right)}\\\ \\ \mathtt{Area\ =\ 18\ \left( x^{1+1}\right)\left( y^{1+2}\right)}\\\ \\ \mathtt{Area=\ 18.\ x^{2} .\ y^{3}}
(04) Find area of right triangle whose base length is \mathtt{16x^{3} y^{3}} and height is \mathtt{x^{5} y^{2}}
Given:
Base of triangle =\mathtt{16x^{3} y^{3}}
Height of Triangle = \mathtt{x^{5} y^{2}}
We know that;
Area of Triangle = \mathtt{\frac{1}{2} \ Base\times \ Height} \\\ \\
\mathtt{Area\ =\frac{1}{2} \ \times 16x^{3} y^{3} \times \ x^{5} y^{2}}\\\ \\ \mathtt{Area=\ \left(\frac{1}{2} .16\right)\left( x^{3} .x^{5}\right)\left( y^{3} .y^{2}\right)}\\\ \\ \mathtt{Area\ =\ 8.\ \left( x^{3+5}\right)\left( y^{3+2}\right)}\\\ \\ \mathtt{Area\ =\ 8\ x^{8} y^{5}}