Question

pls find the pls find​

Answer 1

The example in the given attachment shows the use of associative property of addition in rational numbers. This property says that in any which way we group rational numbers while adding, the sum will remain the same. That is:

a + (b + c) = (a + b) + c

In the question, we are given three rational numbers and two ways in which the addends are grouped. We have to find the sum of both the problems and prove that the associative property of addition surely holds in rational numbers. Let us solve both the ways separately.

$\bf{\bullet \:\:\:\dfrac{-1}{2}+ \Bigg [ \dfrac{3}{7}+ \Big( \dfrac{-4}{3} \Big ) \Bigg ]}$

→ Making the denominators in the brackets same,

$\sf{= \:\dfrac{-1}{2}+ \Bigg [ \dfrac{3\times3}{7\times3}+ \Big ( \dfrac{-4\times7}{3\times7} \Big ) \Bigg ]}$

→ Simplifying,

$\sf{= \:\dfrac{-1}{2}+ \Bigg [ \dfrac{9}{21}+ \Big ( \dfrac{-28}{21} \Big ) \Bigg ]}$

→ Adding the numerators,

$\sf{= \:\dfrac{-1}{2}+ \Bigg [ \dfrac{9+(-28)}{21}\Bigg ]}$

$\sf{= \:\dfrac{-1}{2}+ \dfrac{(-19)}{21}}$

→ Making the denominators same,

$\sf{= \:\dfrac{-1\times21}{2\times21}+ \dfrac{(-19)\times2}{21\times2}}$

→ Simplifying,

$\sf{= \:\dfrac{-21}{42}+ \dfrac{-38}{42}}$

→ Adding the numerators,

$\sf{= \:\dfrac{(-21)+(-38)}{42}}$

$\sf{= \:\dfrac{-59}{42}}$

→ Thus, we get

$\boxed{\:-1\dfrac{17}{42}}$

$\bf{\bullet \:\:\:\Bigg[\dfrac{-1}{2}+\dfrac{3}{7} \Bigg]+ \Big( \dfrac{-4}{3} \Big)}$

→ Making the denominators in the brackets same,

$\sf{=\:\Bigg[\dfrac{-1\times7}{2\times7}+\dfrac{3\times2}{7\times2} \Bigg]+ \Big( \dfrac{-4}{3} \Big)}$

→ Simplifying,

$\sf{=\:\Bigg[\dfrac{-7}{14}+\dfrac{6}{14} \Bigg]+ \Big( \dfrac{-4}{3} \Big)}$

→ Adding the numerators,

$\sf{=\:\Bigg[\dfrac{(-7)+6}{14} \Bigg]+ \Big( \dfrac{-4}{3} \Big)}$

$\sf{=\:\dfrac{-1}{14} + \Big( \dfrac{-4}{3} \Big)}$

→ Making the denominators same,

$\sf{=\:\dfrac{-1\times3}{14\times3} + \dfrac{-4\times14}{3\times14} }$

→ Simplifying,

$\sf{=\:\dfrac{-3}{42} + \dfrac{-56}{42} }$

→ Adding the numerators,

$\sf{=\:\dfrac{(-3)+(-56)}{42}}$

$\sf{=\:\dfrac{-59}{42}}$

→ Thus, we get

$\boxed{\:-1\dfrac{17}{42}}$

In both the ways, the sum remains the same.

$\sf{Therefore,\:\dfrac{-1}{2}+ \Bigg [ \dfrac{3}{7}+ \Big( \dfrac{-4}{3} \Big ) \Bigg ]\:=\:\Bigg[\dfrac{-1}{2}+\dfrac{3}{7} \Bigg]+ \Big( \dfrac{-4}{3} \Big)}$

From this, we can infer that the associative property of addition certainly holds in rational numbers.