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Please could someone explain this to me? (Laws of indices)

I was doing a practice quiz for my exams to do with Laws of indices and I come across the question which is the image below.

In my exam I'm allowed a calculator however when I type the equation in it doesn't give me an answer anywhere near to what is correct.. Could someone explain? Am I typing it in wrong on my calculator, or how do I work it out on paper?

You introduced a negative sign in your answer for no reason

Reply 3

the bear

the negative in the index means "one over"

Reply 4

username5118974

Ohhhhhhh, I understand now thankyou

Reply 5

Tayls102

As someone else mentioned the negative index, x^-nmeans 1/(xⁿ) =(1/x)ⁿ. I think it's important to have an understanding of why this is the case, as it might not be that clear when first seeing this.

Consider powers of 2.

2¹=2, 2²=4, 2³=8, and so on. Each time i increase the index by 1, i multiply by 2. So if i want to go in the other direction i need to divide by 2. so to go from 2³=8 to 2² i divide by 2. ie 2²=(2³)/2=8/2=4.
I can continue in this way to get that 2⁰=(2¹)/2=1 (this is essentially why x⁰=1 for all x).
I can keep decreasing the index by 1, so now 2^-1=(2⁰)/2=1/2, 2^-2=(2^-1)/2=1/4=(1/2)², and so on. In general, we get that 2^-n=(1/2)ⁿ.

If you want the more general result that x^-n=(1/x)ⁿthen you can just replace 2 with x above, the argument is exactly the same.