Dividing monomials

On this post we explain how to do a division of monomials. Also, you will see examples of divisions of monomials and exercises with solutions to practice.

How to divide a monomial by a monomial

In mathematics, the result of the division of two monomials is another monomial whose coefficient is equal to the quotient of the coefficients of the monomials and whose variable is obtained by dividing the variables that have the same base, that is, by subtracting their exponents.

dividing two monomials

Therefore, to divide two different monomials, simply divide the coefficients and subtract the exponents of the powers that have the same base.

Any division of monomials can also be expressed as a fraction:

8x^3y^2z\div 2x^2y=\cfrac{8x^3y^2z}{2x^2y}=4xyz

Finally, you must remember that in the division of the coefficients of monomials the sign rules also apply, since the algebraic division of monomials consists of an arithmetic operation. Therefore:

  • A positive monomial divided by another positive monomial is equal to a positive monomial:

\cfrac{8x^9}{2x^3}=4x^6

  • A positive monomial divided by a negative monomial (or vice versa) is equal to a negative monomial:

\cfrac{-8x^9}{2x^3}=-4x^6

\cfrac{8x^9:}{-2x^3}=-4x^6

  • Dividing two negative monomials results in a positive monomial:

\cfrac{-8x^9}{-2x^3}=4x^6

Dividing monomials is very similar to multiplying monomials, but instead of adding the exponents we have to subtract the exponent of the denominator from the exponent of the numerator.

Examples of divisions of monomials

So that you can fully understand how to divide two or more monomials, below we leave you with several examples of divisions of monomials:

  • \displaystyle\frac{7x^6}{7x^4}=\frac{7}{7}\ x^{6-4}=1x^2=x^2
  • \displaystyle \frac{12y^5}{4y^2}=\frac{12}{4} \ y^{5-2} = 3y^3
  • \displaystyle \frac{15x^7y^6}{3x^4y^5}=\frac{15}{3} \ x^{7-4}y^{6-5} = 5x^3y
  • \displaystyle \frac{27x^9y^7}{-3x^5y^2}= \frac{27}{-3} \ x^{9-5}y^{7-2}= -9x^4y^5
  • \displaystyle -18x^{13}\div 3x^4 \div (-2x^7) = -6x^9\div (-2x^7) = 3x^2

Note that if the dividend monomial has a negative exponent, we actually add the exponents:

  • \displaystyle\frac{10x^4}{5x^{-2}}=\frac{10}{5}\ x^{4-(-2)}=2x^{4+2}=2x^2

Practice problems on dividing monomials

Below you have several practice problems of divisions of monomials to understand better this type of operation with monomials:

Problem 1

Calculate the following divisions of monomials:

\displaystyle\text{A)} \ \frac{24x^4}{6x^2}

\displaystyle\text{B)} \ \frac{16y^9}{-2y^6}

\displaystyle\text{C)} \ \frac{32x^7}{4x^3}

\displaystyle\text{D)} \ \frac{-21a^3}{-3a}

\displaystyle\text{A)} \ \frac{24x^4}{6x^2}=\frac{24}{6} \ x^{4-2}=\bm{4x^2}

\displaystyle\text{B)} \ \frac{16y^9}{-2y^6}=\frac{16}{-2} \ y^{9-6}=\bm{-8y^3}

\displaystyle\text{C)} \ \frac{32x^7}{4x^3}=\frac{32}{4} \ x^{7-3}=\bm{8x^4}

\displaystyle\text{D)} \ \frac{-21a^3}{-3a}=\frac{-21}{-3} \ a^{3-1}=\bm{7a^2}

Note that when a variable has no exponent it means that it is raised to 1. Therefore, in the last operation the term -3a is equivalent to -3a ^ 1 and that is why we must subtract one unit from the exponent of the numerator.

 

Problem 2

Solve the following divisions of monomials:

\displaystyle\text{A)} \ \frac{14x^8y^3}{2x^6y}

\displaystyle\text{B)} \ \frac{45x^{11}y^9z^5}{-5x^6y^2z^3}

\displaystyle\text{C)} \ \frac{-11a^5b^9}{-a^2b^6}

\displaystyle\text{D)} \ \frac{42x^5y^3z^6}{7x^2y^3z^4}

\displaystyle\text{A)} \ \frac{14x^8y^3}{2x^6y}=\bm{7x^2y^2}

\displaystyle\text{B)} \ \frac{45x^{11}y^9z^5}{-5x^6y^2z^3}=\bm{-9x^5y^7z^2}

\displaystyle\text{C)} \ \frac{-11a^5b^9}{-a^2b^6}=\bm{11a^3b^3}

\displaystyle\text{D)} \  \frac{42x^5y^3z^6}{7x^2y^3z^4}= 6x^3y^0z^2=\bm{6x^3z^2}

In the last operation we have simplified the term y^0 because any number raised to 0 is equal to 1. Thus:

 6x^3y^0z^2=6x^3\cdot 1 \cdot z^2=\bm{6x^3z^2}

 

Problem 3

Simplify the following divisions of monomials as much as possible:

 \text{A)} \ 36x^7y^9z^2 \div 6x^2y^4 \div 3x^4y^2z

 \text{B)} \ -50a^{12}b^8c^9 \div (-5a^5b^3c^2) :\div(-2a^4b^2c^4)

 \text{C)} \ 30x^5y^9z^8 \div 2xy^4z^6 \div (-3x^2y^3z)

 \text{D)} \ 48x^8y^6z^{10} :\div(-6x^4y^{2}z^4) \div (-4x^2y^2z^3)

 \text{A)} \ 36x^7y^9z^2 \div 6x^2y^4 \div 3x^4y^2z =  6x^5y^5z^2\div 3x^4y^2z =   \bm{2xy^3z}

 \text{B)} \ -50a^{12}b^8c^9\div (-5a^5b^3c^2) \div (-2a^4b^2c^4) =   10a^7b^5c^7\div (-2a^4b^2c^4) =   \bm{-5a^3b^3c^3}

 \text{C)} \ 30x^5y^9z^8 \div 2xy^4z^6 \div (-3x^2y^3z) =   15x^4y^5z^2\div (-3x^2y^3z) =  \bm{-5x^2y^2z}

 \text{D)} \  48x^8y^6z^{10} \div (-6x^4y^{2}z^4) \div (-4x^2y^2z^3)=  -8x^4y^4z^6\div (-4x^2y^2z^3)=  \bm{2x^2y^2z^3}

 

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