Question

1) Check the commutative property (addition and multiplication) when a = -1/2 b=3/10?


2) Check the associative property of multiplication of rational numbers if a = -2/5 B=3/4 and c= 4/5?


3) If a = -3/5 b=5/7 and c= 3/8
prove that a(b + c) = ab + ac ( Distributive property)?


4) Write the multiplicative inverse and additive inverse of a)-2/7
b)-3/-8
c) 4/-5


5) Multiply by the reciprocal of -8/3?


please answer this question with the correct answers... If answers are correct, will make u brainiest.​

Answer 1

Solution : 1

$\sf \: First \: of \: all \: we \: will \: check \: the \: commutative \: property \: of \: Addition.$

$\boxed{ \sf \: Commutative \: property \: of \: addition = \red{ a + b = b + a }}$

$\sf \implies \: \bigg( \dfrac{ - 1}{2} + \dfrac{3}{10} \bigg) = \bigg( \dfrac{3}{10} + \dfrac{ - 1}{2} \bigg)$

$\implies \sf \: \dfrac{ - 5 + 3}{10} = \dfrac{3+ ( - 5)}{10}$

$\implies \sf \: \dfrac{ -2}{10} = \dfrac{ - 2}{10}$

$\implies \sf \:\dfrac{ -1}{5} = \dfrac{ - 1}{5}$

$\sf \: Now, \: we \: will \: check \: the \: commutative \: property \: of \: Multiplication.$

$\boxed{ \sf \: Commutative \: property \: of \: Multiplication = \red{ a \times b = b \times a }}$

$\sf \implies\bigg( \dfrac{ - 1}{2} \times \dfrac{3}{10} \bigg) = \bigg( \dfrac{3}{10} \times \dfrac{ - 1}{2} \bigg)$

$\implies \sf \:\dfrac{ - 3}{20} = \dfrac{ - 3}{20}$

______________________________________

Solution : 2

$\boxed{\sf \: Associative \: property \: of \: Multiplication = \red{ (a \times b) \times c = a \times (b \times c) }}$

$\sf \implies \: \bigg( \dfrac{ - 2}{5} \times\dfrac{ 3}{4} \bigg) \times \dfrac{4}{5} \: = \: \dfrac{ - 2}{5} \times \bigg(\dfrac{ 3}{4} \times \dfrac{4}{5} \bigg)$

$\sf \implies \: \sf \dfrac{ - 3}{10} \times \dfrac{4}{5} \: = \: \dfrac{ - 2}{5} \times \dfrac{3}{5}$

$\sf \implies \:\sf \dfrac{ - 3}{5} \times \dfrac{2}{5} \: = \: \dfrac{ - 2}{5} \times \dfrac{3}{5}$

$\sf \implies \: \sf \: \dfrac{ - 6}{5} \: = \dfrac{ - 6}{5}$

______________________________________

Solution : 3

$\sf \: Prove \: that \: \underline{a(b + c) = ab + ac} \: (Distributive \: property)$

$\sf \implies \: \sf \: \dfrac{ - 3}{5} \bigg( \dfrac{5}{7} + \dfrac{3}{8} \bigg) = \bigg(\dfrac{ - 3}{5} \times \dfrac{5}{7} \bigg) + \bigg(\dfrac{ - 3}{5} \times \dfrac{3}{8} \bigg)$

$\sf \implies \:\sf \: \dfrac{ - 3}{5} \bigg( \dfrac{5}{7} + \dfrac{3}{8} \bigg) = \sf \: \bigg( \dfrac{ - 15}{35} \bigg) +\bigg( \dfrac{ - 9}{40} \bigg)$

$\sf \implies \:\sf \: \dfrac{ - 3}{5} \times \bigg(\dfrac{40 + 21}{56}\bigg) = \sf \: \dfrac{ (- 15 \times 8) + ( - 9 \times - 7)}{280}$

$\sf \implies \:\sf \: \dfrac{ - 3}{5} \times \dfrac{61}{56} = \sf \:\dfrac{ - 120 - 63}{280}$

$\sf \implies \:\sf \: \dfrac{ - 183}{280} = \dfrac{ - 183}{280}$

______________________________________

Solution : 4

$\sf \: \underline{Multiplicative \: inverse} :$

$\sf \: \implies \: \dfrac{ - 2}{7} = \dfrac{ - 7}{2}$

$\sf \: \implies \: \dfrac{ - 3}{ - 8} = \dfrac{ 8}{3}$

$\sf \: \implies \: \dfrac{4}{ - 5} = \dfrac{ - 5}{ 4}$

$\sf \: \underline{ Additive \: inverse} :$

$\sf \: \implies \: \dfrac{ - 2}{7} = \dfrac{ 2}{7}$

$\sf \: \implies \: \dfrac{ - 3}{ - 8} = \dfrac{ - 3}{ 8}$

$\sf \: \implies \: \dfrac{4}{ - 5} = \dfrac{4}{ 5}$

______________________________________

Solution : 5

$\sf \: \underline{ Correct \: question} :$

$\sf \: Prove \: that : \: \dfrac{ - 8}{3} \times \bigg( \dfrac{ - 3}{8} \bigg) \: = \: 1$

$\sf \implies \: \dfrac{ - \!\!\!\not8}{\!\!\!\not3} \times \dfrac{ - \!\!\!\not3}{\!\!\!\not8} \: = \: 1$

$\sf \implies \: 1 \: = \: 1$

$\sf \: Hence \: proved !$