Square root of (36*45+18*72)

Question

Hello. For the below equation, I factored out 9 as follows. Why is this incorrect?

\sqrt{36*45+18*72} 
 \sqrt{9(4*5+2*8)} 
 \sqrt{9(36)} 
3*6
18

The correct answer is 54.

Thanks!

Bunuel · Answer

You incorrectly factored 9, while you should have factored out 9*9.

\sqrt{36*45+18*72}=\sqrt{(9*4)*(9*5)+(9*2)*(9*8)}=\sqrt{9*9(4*5+2*8)}=9*6=54

Hope it helps.

Kinshook · Answer

\sqrt{36*45+18*72} 
 \sqrt{36(1*45+1*36)} 
 \sqrt{36*81} 
 6*9 = 54

anikafromnyc · Answer

It does. Thank you!

ThrivingWind · Answer

Hi! Do we multiply the 9 twice because we're factoring out of 2 parts of the equation separated by the +? I feel like I've seen equations where it's:

(36*45) + (18*72)
9(4*5) + 9(2*8)
9(20) + 9(16)
9(20+16)
9(36)

How come we can't factor like that?

IanStewart · Answer

If you write out in full what you're doing when going from the first line to the second (the line I've highlighted in red), you might more easily see how we need to factor here. Because 36 = 9*4, we can replace "36" with "9*4", and because 45 = 9*5, we can replace "45" with "9*5". Similarly we can replace 18 with 9*2 and 72 with 9*8. Doing that first, we get:

(36*45) + (18*72) = (9)(4)(9)(5) + (9)(2)(9)(8)

and now since we can multiply numbers in any order, this is equal to

(9)(9)(4)(5) + (9)(9)(2)(8) = (9^2)(4)(5) + (9^2)(2)(8)

and since 9^2 is common to both terms we can factor it out:

9^2

and now we can factor out a '4' if we want to as well.

It's crucial in situations like this to first notice what operation is involved -- the 'rules' for addition are very different from the rules for multiplication. If you're adding (or subtracting), like here:

10 + 45

then we can factor out a 5 by dividing each term by 5 once:

10 + 45 = 5(2 + 9)

but if we are instead multiplying 10*45, and want to factor out a 5, we must only take it from one of the two numbers:

10*45 = (5)(2)(45) = 5 * (2)(45)

or

10*45 = (10)(5)(9) = 5 * (10)(9)

Notice in this case 10*45 is not equal to (5)(2)(9) -- we're missing a '5' if we try to take just one '5' from both numbers in a product. The question in this thread combines addition and multiplication in one problem, so it and similar problems are good practice if you want to confirm you understand how all of this works (something you'll certainly want to understand for the GMAT!).

Square root of (3645+1872)

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