Properties of Rational Numbers

Multiply the $2567424633 \times 8171727272 + 72672289$.

Show that $a÷\left(b÷c\right)\ne \left(a÷b\right)÷c$ for $\frac{2}{3}÷\left(\frac{5}{9}÷\frac{15}{3}\right)\ne \left(\frac{2}{3}÷\frac{5}{9}\right)÷\frac{15}{3}$.

Verify that $\left(a+b\right)+c=a+(b+c)$ where

$$\begin{gathered} \left(a\right) a=\frac{-1}{2}, b=\frac{3}{4}, c=\frac{5}{7} \\ \left(b\right) a=\frac{2}{3}, b=\frac{-3}{4}, c=\frac{7}{5} \end{gathered}$$

Verify Distributive property of multiplication over addition: $p=\frac{2}{3}, q=\frac{3}{4}$ and $r=\frac{5}{6}$.

Integers are non-commutative under division.

Give one example to show that integers are commutative and non-commutative under addition, and division respectively.

Integers are Commutative under addition, and division.

Name the property involved in the following example. $($closure property$/$commutativity$/$associativity$)$

$\frac{5}{2}\times \frac{3}{7}=\frac{15}{14}$

Write the reciprocal of $-5$.

The rational numbers $\frac{1}{2}$ and $\frac{3}{5}$ are commutative with respect to subtraction.

The rational numbers $2$ and $\frac{5}{4}$ are commutative with respect to subtraction.

Find the product of additive inverse and multiplicative inverse of $\frac{-11}{13}$.

Find the sum of additive inverse and multiplicative inverse of $\frac{7}{3}$.

If $\frac{a}{b}$ is the additive inverse of $\frac{c}{d}$, then find the value of $\frac{a}{b}+\frac{c}{d}$.

Find the multiplicative inverse of $312$.

Find the additive inverse of $\frac{17}{-3}$.

Find the additive inverse of $\frac{-6}{-7}$

Give an example to show that subtraction is not associative for rational numbers.

'Rational numbers are commutative under addition but not commutative under subtraction.' Justify the statement with an example.

Find the multiplicative inverse of $\frac{-11}{13}$