Multiplication of Like Algebraic Terms

A mathematical operation of multiplying two or more like algebraic terms is called the multiplication of like algebraic terms.

Introduction

A multiplication sign is displayed between every two like algebraic terms for multiplying the algebraic terms. The product of them is calculated by the product of products of their numerical and literal coefficients. The product of both types of products is written in a row. In this case, the like terms have a literal coefficient commonly. Therefore, the product of literal coefficients of them is equal to the literal coefficient raised to the power of the total number of multiplying terms.

$3ab^2$, $4ab^2$ and $5ab^2$ are three like algebraic terms. The product of multiplication of them can be calculated in four basic simple steps.

First step

Write all the like terms in a row but display a multiplication sign between every two like terms for representing multiplication of the terms.
$3ab^2 \times 4ab^2 \times 5ab^2$

Second step

Write each term in product form as product of numerical and literal coefficients.
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $(3 \times ab^2) \times (4 \times ab^2) \times (5 \times ab^2)$
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $3 \times ab^2 \times 4 \times ab^2 \times 5 \times ab^2$

Third step

Calculate the products of numerical and literal coefficients.
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $3 \times 4 \times 5 \times ab^2 \times ab^2 \times ab^2$
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $(3 \times 4 \times 5) \times (ab^2 \times ab^2 \times ab^2)$
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $60 \times {(ab^2)}^3$

Fourth step

Find the product of the like algebraic terms.
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $60 \times {(a \times b^2)}^3$
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $60 \times \Big({(a)}^3 \times {(b^2)}^3\Big)$
In this case, product rule of exponents is used to simplify the power of an exponential term.
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $60 \times \Big(a^3 \times b^6\Big)$
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $60 \times \Big(a^3b^6\Big)$
$\implies$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $60 \times a^3b^6$
$\,\,\, \therefore \,\,\,\,\,\,$ $3ab^2 \times 4ab^2 \times 5ab^2$ $\,=\,$ $60a^3b^6$

Thus, the product of two or more like algebraic terms is obtained in algebraic mathematics.

Examples

Observe the following examples to understand the process of multiplying like algebraic terms.

$(1)\,\,\,\,\,\,$ $7a \times 5a$ $\,=\,$ $(7 \times 5) \times a^2$ $\,=\,$ $35a^2$

$(2)\,\,\,\,\,\,$ $(-2cd) \times 4cd$ $\,=\,$ $(-2 \times 4) \times {(cd)}^2$ $\,=\,$ $-8c^2d^2$

$(3)\,\,\,\,\,\,$ $3h^2 \times 4h^2 \times 5h^2$ $\,=\,$ $(3 \times 4 \times 5) \times {(h^2)}^3$ $\,=\,$ $60h^6$

$(4)\,\,\,\,\,\,$ $bq^3 \times 2bq^3 \times 3bq^3 \times 4bq^3$ $\,=\,$ $(1 \times 2 \times 3 \times 4) \times {(bq^3)}^4$ $\,=\,$ $24b^4q^{12}$

$(5)\,\,\,\,\,\,$ $4xy^2z^3 \times 5xy^2z^3 \times 6xy^2z^3$ $\,=\,$ $(4 \times 5 \times 6) \times {(xy^2z^3)}^3$ $\,=\,$ $120x^3y^6z^9$

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