# Multiplication of Algebraic Symbols

The system of multiplying one algebraic symbol by another algebraic symbol is called multiplication of algebraic symbols.

### Multiplication of Like Symbols

Like algebraic symbols are involved in multiplication in three different forms and they form exponential form terms as the product of multiplication of the algebraic symbols due to the like property.

1.

#### Like symbols having positive sign

$a$ is a positive algebraic symbol and multiply by the same symbol to perform the multiplication of two or more positive algebraic symbols.

$a×a={a}^{2}$

The like property of the algebraic symbols converts the multiplication of two or more positive like symbols into exponential form as the product.

In the same way,

$a×a×a={a}^{3}$

$a×a×a×a={a}^{4}$

$a×a×a×a×\dots ×a\left(nsymbols\right)={a}^{n}$

2.

#### Like Symbols having Opposite signs

$a$ and $–a$ are two like algebraic symbols which have opposite signs and perform multiplication of both algebraic symbols.

$a×\left(–a\right)$

$⇒a×\left(–a\right)=a×\left(–1\right)×a$

$⇒a×\left(–a\right)=\left(–1\right)×a×a$

$⇒a×\left(–a\right)=\left(–1\right)×{a}^{2}$

Swap the multiplying like symbols.

$\left(–a\right)×a$

$⇒\left(–a\right)×a=\left(–1\right)×a×a$

$⇒\left(–a\right)×a=\left(–1\right)×{a}^{2}$

3.

#### Like symbols having Negative sign

$–a$ is a negative algebraic symbol and multiply it by the same symbol to know the product of negative algebraic symbols.

$\left(–a\right)×\left(–a\right)$

$⇒\left(–a\right)×\left(–a\right)=\left(–1\right)×a×\left(–1\right)×a$

$⇒\left(–a\right)×\left(–a\right)=\left(–1\right)×\left(–1\right)×a×a$

$⇒\left(–a\right)×\left(–a\right)=1×{a}^{2}$

$⇒\left(–a\right)×\left(–a\right)={a}^{2}$

Use same procedure to get product for more than two negative like algebraic symbols.

$\left(–a\right)×\left(–a\right)×\left(–a\right)$

$\left(–a\right)×\left(–a\right)×\left(–a\right)=\left(–1\right)×a\left(–1\right)×a\left(–1\right)×a$

### Multiplication of Unlike Symbols

Two or more unlike algebraic symbols involve in multiplication with same signs or opposite signs. Due to unlike property of the algebraic symbols, they appear in product form as the product of the multiplication of two or more unlike algebraic symbols.

1.

#### Multiplication of Positive Unlike Symbols

$a$ and $b$ are two alphabetic letters and also two positive algebraic symbols. Multiply them and same symbols are appeared in product form as the product.

$a×b=ab$

Same pattern is repeated for the multiplication of more than two unlike algebraic symbols.

$a×b×c=abc$

$a×b×c×d=abcd$

$a×b×c×d×\dots ×n=abcd\dots n$

In other words, if two or more positive unlike algebraic symbols participate in multiplication, the same symbols are appeared in product form as the product of multiplication.

2.

#### Multiplication of Opposite sign Unlike Symbols

$a$ and $–b$ are two unlike algebraic symbols which have opposite signs and multiply them.

$a×\left(–b\right)$

$⇒a×\left(–b\right)=a×\left(–1\right)×b$

$⇒a×\left(–b\right)=\left(–1\right)×a×b$

$⇒a×\left(–b\right)=\left(–1\right)×ab$

Now, multiply $–a$ and $b$. They are also unlike algebraic symbols which have opposite signs.

$\left(–a\right)×b$

$⇒\left(–a\right)×b=\left(–1\right)×a×b$

$⇒\left(–a\right)×b=\left(–1\right)×ab$

If two unlike algebraic symbols with opposite signs involve in multiplication, the unlike algebraic symbols appear with negative sign in product form as the product of multiplication.

3.

#### Multiplication of Negative Sign Unlike Symbols

$–a$ and $–b$ are two unlike negative algebraic symbols and multiply them for the product.

$\left(–a\right)×\left(–b\right)$

$⇒\left(–a\right)×\left(–b\right)=\left(–1\right)×a×\left(–1\right)×b$

(or)

$⇒\left(–a\right)×\left(–b\right)=1×a×b$

$⇒\left(–a\right)×\left(–b\right)=1×ab$

$⇒\left(–a\right)×\left(–b\right)=ab$

Consider three negative unlike algebraic symbols and perform multiplication.

$\left(–a\right)×\left(–b\right)×\left(–c\right)$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)=\left(–1\right)×a×\left(–1\right)×b×\left(–1\right)×c$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)=\left(–1\right)×\left(–1\right)×\left(–1\right)×a××b×c$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)={\left(–1\right)}^{3}×abc$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)=\left(–1\right)×abc$

Use four negative unlike algebraic symbols to perform multiplication.

$\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)=\left(–1\right)×a×\left(–1\right)×b×\left(–1\right)×c×\left(–1\right)×d$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)=\left(–1\right)×\left(–1\right)×\left(–1\right)×\left(–1\right)×a××b×c×d$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)={\left(–1\right)}^{4}×abcd$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)=1×abcd$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)=abcd$

Perform multiplication with five negative algebraic symbols in same procedure.

$\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)×\left(–e\right)$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)×\left(–e\right)=\left(–1\right)×a×\left(–1\right)×b×\left(–1\right)×c×\left(–1\right)×d×\left(–1\right)×e$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)×\left(–e\right)=\left(–1\right)×\left(–1\right)×\left(–1\right)×\left(–1\right)×\left(–1\right)×a××b×c×d×e$

$⇒\left(–a\right)×\left(–b\right)×\left(–c\right)×\left(–d\right)×\left(–e\right)={\left(–1\right)}^{5}×abcde$

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