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Multiplication of Unlike Algebraic Terms

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

Introduction

A multiplication sign is displayed between every two unlike algebraic terms for representing the multiplication of unlike terms in mathematical form. The product of unlike terms is actually calculated by the product of products of their numerical and literal coefficients. In this case, the literal coefficients of them are different but they may have one or more literals with same or different powers.

$2a$, $3a^2b$ and $4ab^2$ are three unlike algebraic terms. The product of multiplication for unlike algebraic terms can be calculated in four simple steps.

First step

Write all the unlike terms in a row but display a multiplication sign between every two unlike terms for expressing multiplication of the terms in mathematical form.
$2a \times 3a^2b \times 4ab^2$

Second step

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

Third step

Calculate the products of numerical and literal coefficients. If literal coefficients of two or more unlike terms are formed by one or more literals, then the product of literal coefficients are simplified by the product rule of exponents.
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $2 \times 3 \times 4 \times a \times a^2b \times ab^2$
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $(2 \times 3 \times 4) \times (a \times a^2b \times ab^2)$
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $24 \times (a \times a^2b \times ab^2)$
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $24 \times (a \times a^2 \times b \times a \times b^2)$
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $24 \times (a \times a^2 \times a \times b \times b^2)$
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $24 \times (a^{1+2+1} \times b^{1+2})$
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $24 \times (a^4 \times b^3)$
$\implies$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $24 \times (a^4b^3)$

Fourth step

Find the product of the unlike algebraic terms.
$\,\,\, \therefore \,\,\,\,\,\,$ $2a \times 3a^2b \times 4ab^2$ $\,=\,$ $24a^4b^3$

In this way, the multiplication of two or more unlike algebraic terms are calculated algebraically in mathematics.

Examples

Look at the following examples to understand the procedure for multiplying unlike algebraic terms.

$(1)\,\,\,\,\,\,$ $3a \times 4b$ $\,=\,$ $(3 \times 4) \times (a \times b)$ $\,=\,$ $12ab$

$(2)\,\,\,\,\,\,$ $2c^2d \times 5cde$ $\,=\,$ $(2 \times 5) \times (c^2d \times cde)$ $\,=\,$ $10c^3d^2e$

$(3)\,\,\,\,\,\,$ $(-6h) \times 7ij \times 5k^2$ $\,=\,$ $(-6 \times 7 \times 5) \times (h \times ij \times k^2)$ $\,=\,$ $-210hijk^2$

$(4)\,\,\,\,\,\,$ $3p^2 \times q \times 6r^3 \times 7s^5$ $\,=\,$ $(3 \times 1 \times 6 \times 7) \times (p^2 \times q \times r^3 \times s^5)$ $\,=\,$ $126p^2qr^3s^5$

$(5)\,\,\,\,\,\,$ $9ax \times 10by \times 6cz$ $\,=\,$ $(9 \times 10 \times 6) \times (ax \times by \times cz)$ $\,=\,$ $540abcxyz$

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