UPSKILL MATH PLUS

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### Theory:

The distributive property is one of the most frequently used properties in mathematics. This property lets you multiply each addend separately and then add the products.
Now let's see the distributive property of multiplication over addition.
If $$a$$, $$b$$ and $$c$$ are three integers, then:

$$a ×$$ ($$b + c$$) $$=$$ ($$a × b$$) $$+$$ ($$a × c$$)

Let $$a =$$ $\frac{-8}{2}$, $$b =$$ $\frac{-4}{4}$, and $$c =$$ $\frac{1}{3}$.

Then, $\frac{-8}{2}×\left(\frac{-4}{4}+\frac{1}{3}\right)=\left(\frac{-8}{2}×\frac{-4}{4}\right)+\left(\frac{-8}{2}×\frac{1}{3}\right)$

First, we simplify the LHS expression. Since denominators are not the same, take LCM to add the numbers.

$\frac{-8}{2}×\left(\frac{-4}{4}+\frac{1}{3}\right)$

LCM of 4 and 3 is $$12$$.

$\begin{array}{l}=\frac{-8}{2}×\left(\frac{-4×3}{4×3}+\frac{1×4}{3×4}\right)\\ \\ =\frac{-8}{2}×\left(\frac{-12}{12}+\frac{4}{12}\right)\\ \\ =\frac{-8}{2}×\left(\frac{-12+4}{12}\right)\\ \\ =\frac{-8}{2}×\frac{-8}{12}\\ \\ =\frac{64}{2×12}=\frac{64}{24}\end{array}$

Similarly, if we simplify the RHS, we get the same answer.

$\begin{array}{l}=\left(\frac{-8}{2}×\frac{-4}{4}\right)+\left(\frac{-8}{2}×\frac{1}{3}\right)\\ \\ =\left(\frac{32}{2×4}\right)+\left(\frac{-8}{2×3}\right)\\ \\ =\left(\frac{32}{8}+\frac{-8}{6}\right)\end{array}$

Taking LCM

$\begin{array}{l}=\left(\frac{32×3}{8×3}+\frac{\left(-8\right)×4}{6×4}\right)\\ \\ =\left(\frac{96+\left(-32\right)}{24}\right)\\ \\ =\frac{64}{24}\end{array}$

Hence, $$LHS = RHS$$, it shows that $\frac{-8}{2}×\left(\frac{-4}{4}+\frac{1}{3}\right)=\left(\frac{-8}{2}×\frac{-4}{4}\right)+\left(\frac{-8}{2}×\frac{1}{3}\right)$

Therefore multiplication is distributive over addition for rational numbers $$Q$$.