Thanks

@the-blade-dancer So sorry for the late response! Basically, the sentence explains why we think that 162 * r^8 = 32/81. It's essentially an educated guess...

As the solution says, the denominator of 32/81 is 3^4. This is the same thing as saying the number has a factor of 3^-4 (because 32/81 = 32 * 1/81 = 32 * (1/3^4) = 32 * 3^-4). On the other hand, 162 has a factor of 3^4 = 81. So to get from 162 to 32/81, thinking purely in terms of the powers of 3, we are trying to get from

3^4 * something to 3^(-4) * something

Well, to do that, we have to divide by 3^8. We just found that the ratio is sqrt(2)/3, so we want to multiply by the ratio 8 times: this gives (sqrt(2))^8 / 3^8 so it's like dividing by 3^8, which is what we want.

I hope that makes sense!

Why were the two 4's not eating lunch?
Because they already 8!

Why were 3 and 5 so scary?
Because they 8 someone!

Fortnite, Roblox, Minecraft. Those are my 3 main games, I mostly play Fortnite but if get bored then Roblox and if I get bored of Roblox, I play Minecraft.

The number zero is neither positive nor negative. Positive and negative numbers are sometimes called signed numbers.

@quacker88 Ok thank you I got that

Great question, @the-blade-dancer !

If you remember the binomial theorem at all from Module 3 (let me know if you need more explanation here), but we can use a shortcut for the expansion of \((\sqrt{a}+\sqrt{b})^3\).

In general, \((x+y)^3=x^3+3x^2y+3xy^2+y^3\), so

\((\sqrt{a}+\sqrt{b})^3=a\sqrt a +3a\sqrt b+ 3\sqrt a b+b\sqrt b\)
\((\sqrt{a}+\sqrt{b})^3= (a+3b)\sqrt a+(3a+b)\sqrt b\)

This gets us straight to the form that we need. And since this has to be equal to \(22\sqrt 7+26\sqrt 5\), it is most likely that \(a=7, b=5\), so we can go and check that out. After confirming that it is true, we get that our answer is \(7+5=\boxed{12}\). Let me know if you have any other questions!

@perceptiveeagle Oh that.

@quacker88 Thanks for your help!