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

The like algebraic terms have a literal factor commonly. When an algebraic term is divided by its like algebraic term, then the literal factors of both like algebraic terms get cancelled but quotient of them is equal to a number and it is the quotient of both numerical factors of them.

$4xy^2$ and $-6xy^2$ are two like algebraic terms.

01

Take, the algebraic term $4xy^2$ is divided by its like algebraic term $-6xy^2$ and it is written in mathematical form as follows.

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

02

An algebraic term is a product of a numerical and a literal factor. So, each term can be expressed in product of the factors.

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

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

In this case, the literal factor of both terms is same and it is $xy^2$. Hence, they both get cancelled.

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

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

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

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

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

The example has proved that the quotient of division of any two like algebraic terms is always a number. Hence, just find the quotient of numerical factors of the like terms when two like algebraic terms involve in division.

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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