Question

Verify the associative law of multiplication for the rational numbers -(3/13),-(5/7),9/23 also verify the distributive property of addition over multiplication​

Answer 1

GIVEN :

The rational numbers are $-\frac{3}{13}$, $-\frac{5}{7}$ , $\frac{9}{23}$

TO VERIFY :

The associative law of multiplication for the given rational numbers and also verify the distributive property of addition over multiplication​.

SOLUTION :

Given rational numbers are $-\frac{3}{13}$, $-\frac{5}{7}$ , $\frac{9}{23}$

For any rational numbers a, b and c:

The Associative law over multiplication is given by

$a\times (b\times c)=(a\times b)\times (a\times c)$

Let $a=-\frac{3}{13}$, $b=-\frac{5}{7}$ , $c=\frac{9}{23}$

Now substituting the values in the formula,

Now verify the Associative law over multiplication

Now taking LHS

$a\times (b\times c)$

$-\frac{3}{13}\times (-\frac{5}{7}\times \frac{9}{23})$

$=-\frac{3}{13}\times (\frac{-45}{161})$

$=\frac{135}{2093}$

$-\frac{3}{13}\times (-\frac{5}{7})\times \frac{9}{23})=\frac{12}{161}=0.0645$=LHS

Now RHS $(a\times b)\times c$

$(-\frac{3}{13}\times (-\frac{5}{7}))\times \frac{9}{23}$

$=\frac{15}{91}\times \frac{9}{23}$

$=\frac{135}{2093}$

=0.0645=RHS

∴ LHS = RHS

The Associative law over multiplication for the given rational numbers $-\frac{3}{13}$, $-\frac{5}{7}$ , $\frac{9}{23}$ is verified.

∴ $-\frac{3}{13}\times (-\frac{5}{7}\times \frac{9}{23})=((-\frac{3}{13})\times (-\frac{5}{7}))\times \frac{9}{23}$

For any rational numbers a, b and c:

The Distributive property of addition over multiplication is given by

$a\times (b+c)=a\times b+a\times c$

Let $a=-\frac{3}{13}$, $b=-\frac{5}{7}$ , $c=\frac{9}{23}$

Now substituting the values in the formula,

Now verify the Distributive property of addition over multiplication

Now taking LHS

$a\times (b+c)$

$-\frac{3}{13}\times (-\frac{5}{7}+\frac{9}{23})$

$=-\frac{3}{13}\times (\frac{-115+63}{161})$

$=-\frac{3}{13}\times (\frac{-52}{161})$

$=\frac{12}{161}$

$-\frac{3}{13}\times (-\frac{5}{7}+\frac{9}{23})=\frac{12}{161}=0.0745$=LHS

Now RHS $a\times b+a\times c$

$-\frac{3}{13}\times (-\frac{5}{7})+(-\frac{3}{13})\times \frac{9}{23}$

$=\frac{15}{91}-\frac{27}{299}$

$=\frac{4485-2457}{27209}$

$=\frac{2028}{27209}$

=0.0745=RHS

∴ LHS = RHS

The Distributive property of addition over multiplication for the given rational numbers $-\frac{3}{13}$, $-\frac{5}{7}$ , $\frac{9}{23}$ is verified.

∴ $-\frac{3}{13}\times (-\frac{5}{7}+\frac{9}{23})=-\frac{3}{13}\times (-\frac{5}{7})+(-\frac{3}{13})\times \frac{9}{23}$

Answer 2

Step-by-step explanation:

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