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#32 on page 156 has a rather simple sqrt problem, which I am able to solve using a brute force method for finding the sqrt. However, I'd like to understand the book's solution for finding the solution more quickly, but don't quite follow it. Any help would be appreciated.

I've attached an image showing the book's solution. I'm wondering how I am able to drop the 16 as one of the root multiples of 256 (the product of 8*32).

Appreciate any help!

Attachments

sqrt_problem.png [ 182.14 KiB | Viewed 4211 times ]

The solution in book is quite strange. Generally \(\sqrt{a+b}\), DOES NOT equal to \(\sqrt{a}+\sqrt{b}\). Example: \(\sqrt{16+16}=\sqrt{32}=4*\sqrt{2}\) and NOT \(\sqrt{16}+\sqrt{16}=4+4=8\).

Back to the question:

As I understand the problem is:

\(\sqrt{16*20+8*32}\) if yes then: \(\sqrt{16*(20+8*2)}=\sqrt{16*36}=4*6=24\)

Argh. Sorry, there is a typo in my image - you are correct that it should be \(\sqrt{16}*\sqrt{36}\) and not \(\sqrt{16}+\sqrt{36}\).

I am embarrassed that I still do not understand how (8*32) becomes (8*2)? 8*32 is 256, the root of which is 16. So by my thinking, that would leave me with \(\sqrt{(16)(20)+(16)(16)}\). How do you get \(\sqrt{(16)(20)+(8)(2)}\) from that?

Argh. Sorry, there is a typo in my image - you are correct that it should be \(\sqrt{16}*\sqrt{36}\) and not \(\sqrt{16}+\sqrt{36}\).

I am embarrassed that I still do not understand how (8*32) becomes (8*2)? 8*32 is 256, the root of which is 16. So by my thinking, that would leave me with \(\sqrt{(16)(20)+(16)(16)}\). How do you get \(\sqrt{(16)(20)+(8)(2)}\) from that?

Thanks again.

The red part is not correct we'll get not the expression you wrote but: \(\sqrt{16*(20+8*2)}\). Look at the brackets.

And here is how:

Let's just forget about the square root for a moment. We have:

\(16*20+8*32\) no need to multiply \(8\) by \(32\), it's a long way, though still correct. Just take \(16\) from the brackets:

sqrt{ 16 (20 + 16)} okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.

The point is that we are not adding square root of 20 and square root of 16, we are adding 20 and 16 under the SAME square root. Below it he solution to the problem from OG:

\(\sqrt{16*20+8*32}=?\) --> factor out 16: \(\sqrt{16(20+8*2)}=\sqrt{16*36}=4*6=24\).

sqrt{ 16 (20 + 16)} okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.

The point is that we are not adding square root of 20 and square root of 16, we are adding 20 and 16 under the SAME square root. Below it he solution to the problem from OG:

\(\sqrt{16*20+8*32}=?\) --> factor out 16: \(\sqrt{16(20+8*2)}=\sqrt{16*36}=4*6=24\).

I just have one question, wouldn't there still be a 1 left over when you factor out the 16? I'm just thinking if you were to reverse the operation, what would the 16 get multiplied into?
_________________

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sqrt{ 16 (20 + 16)} okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.

The point is that we are not adding square root of 20 and square root of 16, we are adding 20 and 16 under the SAME square root. Below it he solution to the problem from OG:

\(\sqrt{16*20+8*32}=?\) --> factor out 16: \(\sqrt{16(20+8*2)}=\sqrt{16*36}=4*6=24\).

I just have one question, wouldn't there still be a 1 left over when you factor out the 16? I'm just thinking if you were to reverse the operation, what would the 16 get multiplied into?

You mean it should be \(\sqrt{16(20*1+8*2)}=\sqrt{16*36}=4*6=24\)? Yes, but 20*1 = 20, so we can omit 1 there.

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