Is this a graceful method of solving this exponent problem?

[cupid_stunt]

[math]\frac{22^2 - 17^2}{20^2-19^2}[/math]rather than just evaluating by hand?

 

[Otis]

evaluating by hand

Hi. Have you learned the special factoring pattern for a difference of squares? Using that pattern to factor the numerator and denominator will lead to a nice cancellation.

(a^2 - b^2) = (a + b)(a - b)

 

[pka]

Grace is in the brain of the the beholder.
I might do this as [imath](22^2-17^2)=(22+17)(22-17)=(39)(5)=195[/imath]

 

[Otis]

Grace in in the brain...
(39)(5)=195

That'll work, eventually leading to the final answer.

As an alternative, factoring both the numerator and denominator leads directly to the final answer (without having to multiply anything).

 

[Steven G]

No, you should evaluate by hand! I'm not saying that you should square anything. To compute, for example, the denominator 20² - 19² all you need to realize is that it equals (20-19)(20+19) = 1*39 = 39.

I think that it is best to hold off with multiplication since maybe things will cancel out. I know that I did multiply 1 and 39 but that is just 39.
For the numerator, I would not have multiply the 5 and 39 just incase I could reduce it with the denominator. And that is exactly what happened!

 

[Otis]

No, you should evaluate by hand!

Yes, using paper and pencil! Except (possibly) for the arithmetic.

I did multiply 1 and 39

Betcha you wouldn't have, had you written the factorized ratio on paper.

And, you likely would have done all of the arithmetic mentally -- additions and subtractions -- if I know you, which I kinda do, because it takes one to know one!!

?

 

[lookagain]

[math]\frac{22^2 - 17^2}{20^2-19^2}[/math]rather than just evaluating by hand?

Your question confused me because you mistakenly have the word "this" in your heading instead of "there."
That is, you were not showing any method. It is understood you will work it "by hand."

cupid_stunt, a safer phrasing you might use is this:

What is "one of the most efficient ways" to solve this exponent problem without squaring every number?

Posts #2, #4, and #5, so far dealt with this.

Sometimes "efficient ways" overlap with what are sometimes called "elegant" ways.

 

[Only_Vibez]

Wait wait wait. Just wondering here. How would that even matter? Like, yes, good handwriting is always a key thing to have, but. I'm just confused I guess is what i'm trying to say.

 

[mmm4444bot]

How would that even matter?

What are you asking about? Please be specific.

I'm just confused I guess is what i'm trying to say.

What are you confused about? People can't read your mind, right?

 

[Only_Vibez]

How would effectiveness be a problem occurring in elegentness...? Does that make sense. I'm so confused.

 

[lookagain]

How would effectiveness be a problem occurring in elegentness...? Does that make sense. I'm so confused.

I skimmed through the posts three times, but I did not see "effectiveness." Are you intending the word
"efficient?"