C4 biomial expansion help pls

how do I solve section b in C4 june 2018 Q1

Cheeky little blighter this one!!

You know that setting 4 - 9x = 310 won't give a valid value for x.

But, if you note that 310 = 100 x 3.1, then sqrt(310) = 10sqrt(3.1)

Setting 4 - 9x = 3.1 will give you a valid and somewhat simple value to shove into your binomial expansion.

but how do you know what to divide by?

and what if it was cubic like this?

It doesn't matter what the power is. In this question, a cube root means the power will be 1/3, which is what you'll shove in as your value for n in the expansion.

As for your other question of "what to divide by", I'm sure that someone will be along to quote some formula or rule.

Personally, I'm more inclined to look at the process. Taking 10 outside a square root seems sensible since you're making the original number inside it 100 times smaller. For a cube root, taking 10 out will make the original number 1000 times smaller.

But I doubt that you could apply this as a general rule. It may be the case that taking 2 outside a square root and making the original number 4 times smaller, or taking 5 out of a cube root to make the number 125 times smaller might be the way forward.

Hopefully the examiners will keep it to taking 10 out!!

just one more question
after getting 10(1.3)^0.5=1.76234375 what should we do? in the ms i sais that we multiply by 10 but I don't get why honestly

Just to clarify that you've entered a typo - you've written 10(1.3)^0.5=1.76234375, but did you mean (and actually calculated) 10(3.1)^0.5=1.76234375, using x=0.1?

I'm awaiting a D'Oh in response

I suspect that you've actually calculated sqrt(3.1) correctly as 1.762..., but have forgotten that the answer is 10 times this.

Bit odd, since you actually typed "10(3.1)^0.5" (I've corrected your typo), but obviously forgot to multiply by 10.

I'm sorry I'm still lost so once I finish I have to multiply by whatever I got out of the root?

Yep. They are asking you to find the value of sqrt(310), which can't be done directly in the expansion.

But sqrt(310) =10 × sqrt(3.1), which can be done.

You have found our sqrt(3.1) correctly, so you need to multiply it by 10 to find sqrt(310)