prove (ā+b)×(ā-b) = 2(b×ā)
Answers
Explanation:
We can show that the RHS is equal to the LHS by expanding and simplifying.
When expanding two expressions, it is important to use FOIL; First, Outer, Inner, Last. This is the ideal order in which you can multiply each term in each expression by each term in the other expression. Let’s see how this works.
So, First means we multiply the first terms of each expression together, in this case they are both [math]a[/math], so their product is [math]a^2[/math].
Outer means we multiply the terms on the extreme end of the entire product(the first term of the first expression and the last term of the second expression) which is [math]a[/math] and [math]-b[/math], so their product is [math]-ab[/math].
Inner means we multiply the last term of the first expression and the first term of the second expression, which is [math]b[/math] and [math]a[/math], so their product is [math]ba[/math], which is equivalent to [math]ab[/math] by the commutative property of multiplication.
Finally, Last means we multiply the last terms of each expression, which are [math]b[/math] and [math]-b[/math], so their product is [math]-b^2[/math].
Now, we can finally put everything together;
[math](a+b)(a-b) [/math]
[math]= a^2 - ab + ba + (-b^2)[/math]
[math]= a^2 - ab + ab - b^2[/math]
[math]= a^2 - b^2[/math]
Therefore, by expanding and simplifying, we have shown that
[math]a^2 - b^2 = (a+b)(a-b)[/math].
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