Question

Find 17 \[\times \]23 by using distributive property.

Answer 1

Hint: Write 17 and 23 in terms of any two numbers as sum or difference and then use distributive property.
a.(b+c) = a.b + a.c

Complete step-by-step answer:
Distributive property means, if we multiply the sum of two or more addends by a number it will give the same result as multiplying each addend separately and then adding their products together.
 We will find the number which lies exactly in the middle of 17, 23 as the process of distribution becomes easy.
So let the number be N.
N = average of 17 and 23 = \[\dfrac{17+23}{2}\] = 20
N = 20.
So the difference between 17 and 20 is the same as the difference between 20 and 23.
Now we can write 17 as 20 - 3,
We can write 23 as 20 + 3.
Now the algebraic expression 17 \[\times \]23 can be written as:
= (20 - 3) \[\times \](20 + 3)
Assume (20+3) as one entity as apply distributive law,
a.(b+c) = a.b + b.c
we get:
= 20\[\times \](20 + 3) - 3\[\times \](20 + 3)
Now by applying distributive law again in each term, we get:
= \[20\times 20\text{ }+\text{ }20\times 3\text{ }-\text{ }3\times 20\text{ }-\text{ }3\times 3\]
Cancelling the term 20\[\times \]3 we get
= \[20\times 20\text{ }-\text{ }3\times 3\]
By simplifying, we get:
= 400 – 9
= 391
So, we conclude by saying 17 \[\times \]23 = 391.
\[\therefore \]By distributive property we proved 17 \[\times \]23 = 391.

Note: There is a shortcut, in between when we got the expression, (20 - 3) \[\times \](20 + 3)
We can apply (a + b).(a – b) = \[{{a}^{2}}-{{b}^{2}}\], which leads us directly to the last step.