Number Properties (Exponents + Roots)

[iil-london]

sqrt((16)(20)+(8)(32)) =

(A) 4 sqrt(20)
(B) 24
(C) 25
(D) 4 sqrt(20) + 8 sqrt(2)
(E) 32

What is the best way to answer this question as quickly and efficiently as possible ?

Thanks.

[Rahul]

I solved the problem as follows:

=sqrt (16(20+(8)(2))
=sqrt(16(20+16))
=sqrt(16*36)
=4*6
=24

[rfernandez]

Nice solution, Rahul!

The key to Rahul's approach is to identify perfect square factors that you can "pull out."

Another way to get there is to convert everything to prime factors. Depending on your preferences this may or may not be seen as "the best way to answer" it, as it does require a few more steps.

sqrt[(16)(20)+(8)(32)] =
sqrt[(2^4*2^2*5)+(2^3*2^5)]
sqrt[2^6*5+2^8]
sqrt[2^6(5+2^2)]
2^3*sqrt[5+2^2]
8*sqrt(9)
24

Rey

[ElizabethS105]

Hi,

Instead of pulling out a 16, I just pulled out a 4 to get sq.root of 4(4*5) + 4(2*8) --> sq.root of 4(20+16) --> sq.root of 4 * sq.root of 36 --> 12
Can you please tell me where I made a mistake? I don't see why I couldn't pull the 4 out.

Thanks!

[RonPurewal]

here, the problem has nothing to do with the square root. the problem is what you're doing with the numbers inside it.

you can actually find your own mistake here—by actually finding the numbers at each step of your process.

first step:
16*20 + 8*32 --> this is 320 + 256 = 576.

second step:
4(4*5) + 4(2*8) --> this is 4(20) + 4(16)
= 80 + 64
= 144 = WHOA that's a lot smaller... something's wrong here.

now you know the exact step where something went wrong. that should make it much easier for you to figure out the problem.


try to find the problem on your own!

i'll explain it below—but, really, don't just scroll down and be like 'uh-huh, okay, yeah'. try to actually find the mistake yourself first, now that the problem has had a cooling-off period.

[RonPurewal]

so:

the problem is that you're trying to 'get fancy' with multiplication... and you're forgetting that, essentially, multiplication is multiplication is multiplication.

e.g., let's look at the first half of the expression:
16 x 20
= (4 x 4) x (4 x 5)
...and that's what it is. you can multiply these things in any order you want (or you can break them down further—e.g., 2 x 2 x 2 x 2 x 2 x 2 x 5). but you can't do anything fancier than that.

you're trying to pull a '4' out of both sets of parentheses.
if you like fancy words, you're trying to use a 'distributive property'.
the problem is that this property only exists when you're 'distributing over' addition and/or subtraction. e.g., 4(x + y) is actually 4x + 4y, and, likewise, 4(x – y) is actually 4x – 4y.

4(xy), on the other hand, is just, well, 4xy (NOT 4x times 4y).
in that case, since it's all multiplication, the parentheses don't even really do anything (hence why you don't normally write them).

[RonPurewal]

in any case, there are two big takeaways you should get here.

1/
don't let numbers stop being numbers!
this kind of mistake (trying to 'distribute' multiplication over another multiplication) can only happen if you've allowed the numbers to devolve into essentially meaningless symbols that you're pushing around according to completely arbitrary rules.
if, at each step, you remember that the numbers are numbers—i.e., things with values that you can check—then you'll have a safeguard against incorrect operations. WHOA different number... better check that.

incidentally, this is why algebra is inherently harder than arithmetic, especially when it comes to word problems—because, unlike numbers, algebraic symbols really are 'meaningless symbols that we're pushing around'. (the rules all have justifications, of course, but it's impossible to do algebra in real time if you have to remember why every single step works or doesn't work.)
so, basically, if you do have the blessing of working with concrete numbers... take advantage!

[RonPurewal]

and...

2/
if some rule is problematic, try to 'anchor' it in something else that you know.

see, there are lots and lots of common threads that run through mathematics. if thing X is elusive, then there's probably some thing Y, to which it's related, with which you have much less trouble.

e.g.,
i'm guessing here, but i bet that you understand the whole idea of prime factorizations—the idea that you can break up any whole number (>1) into primes, and, specifically, that there's only one way to do so.

if you understand this, then you have another way to understand why you can't write 4 x (5 x 6) = (4 x 5) x (4 x 6)... if you could, then you'd have two different factorizations! woops.

[ElizabethS105]

You nailed it, thank you.

[tim]

[RonPurewal]

You nailed it, thank you.

no problem.