A mathematical operation of dividing an algebraic term by its like term is called the division of like algebraic terms.

The division of any two like algebraic terms is expressed by displaying a division sign between them for dividing them. The quotient of the like algebraic terms is calculated by only finding the quotient of the numerical coefficients due to the same literal coefficient. Actually, the same literal coefficients of both like terms are cancelled in division. Hence, the literal coefficient can be ignored while doing the division with like terms in algebra.

$4xy^2$ and $-6xy^2$ are two like algebraic terms. Divide the term $4xy^2$ by $-6xy^2$ to find their quotient.

Express the division of the like algebraic terms in mathematical form.

$4xy^2 \div (-6xy^2)$

$\implies$ $\dfrac{4xy^2}{-6xy^2}$

Factorize the each algebraic term as its numerical and literal coefficients.

$\implies \dfrac{4xy^2}{-6xy^2} \,=\, \dfrac{4 \times xy^2}{-6 \times xy^2}$

$\implies \dfrac{4xy^2}{-6xy^2} \,=\, \dfrac{4}{-6} \times \dfrac{xy^2}{xy^2}$

Find the quotient of the like algebraic terms by cancelling the literal coefficients, and then finding the quotient of the numerical coefficients.

$\require{cancel} \implies \dfrac{4xy^2}{-6xy^2} \,=\, \dfrac{4}{-6} \times \dfrac{\cancel{xy^2}}{\cancel{xy^2}}$

$\implies \dfrac{4xy^2}{-6xy^2} \,=\, \dfrac{4}{-6} \times 1$

$\implies \dfrac{4xy^2}{-6xy^2} \,=\, -\dfrac{4}{6}$

$\implies \require{cancel} \dfrac{4xy^2}{-6xy^2} \,=\, -\dfrac{\cancel{4}}{\cancel{6}}$

$\therefore \,\,\,\,\,\, \dfrac{4xy^2}{-6xy^2} \,=\, -\dfrac{2}{3}$

Thus, the division of any two like algebraic terms can be calculated in algebraic mathematics. Remember, the quotient of any two like terms is a rational number.

Observe the following examples to understand how to divide an algebraic term by its like term.

$(1) \,\,\,\,\,\,$ $\dfrac{a}{7a}$ $\,=\,$ $\require{cancel} \dfrac{1 \times \cancel{a}}{7 \times \cancel{a}}$ $\,=\,$ $\dfrac{1}{7}$

$(2) \,\,\,\,\,\,$ $\dfrac{3b^2}{6b^2}$ $\,=\,$ $\require{cancel} \dfrac{\cancel{3} \times \cancel{b^2}}{\cancel{6} \times \cancel{b^2}}$ $\,=\,$ $\dfrac{1}{2}$

$(3) \,\,\,\,\,\,$ $\dfrac{25cd^3}{cd^3}$ $\,=\,$ $\require{cancel} \dfrac{25 \times \cancel{cd^3}}{1 \times \cancel{cd^3}}$ $\,=\, 25$

$(4) \,\,\,\,\,\,$ $\dfrac{0.1e^2f^2}{0.01e^2f^2}$ $\,=\,$ $\require{cancel} \dfrac{\cancel{0.1} \times \cancel{e^2f^2}}{\cancel{0.01} \times \cancel{e^2f^2}}$ $\,=\, 10$

$(5) \,\,\,\,\,\,$ $\dfrac{56gh^2i^3j^4}{7gh^2i^3j^4}$ $\,=\,$ $\dfrac{\cancel{56} \times \cancel{gh^2i^3j^4}}{\cancel{7} \times \cancel{gh^2i^3j^4}}$ $\,=\, 8$

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