Yesterday’s post built up some of the things we need to answer a question where the power is negative and a fraction.

One key point is that for all numbers n

n^{0} = 1 and n^{1} = n.

With negative powers we still need to maintain the rule of adding powers.

2^{x} x 2^{-x} = 2^{x-x} (and since x – x x = 0) = 2^{0} = 1

Re-arrange this and we get 2^{-x} = 1/2^{x}

And there the first new rule – a negative power is 1/the positive power

3^{2} = 9 so 3^{-2} = 1/9

To get the second rule we need to consider how powers can be combined.

(n^{2})^{3} = n^{6} – When you raise a power to a power – multiply the powers

[Consider n x n x n x n x n x n]

Now look at

(n^{2})^{1/2} = n^{1} = n

So what does raise to power of half mean if this involves getting from n^{2} to n? Taking the square root! We could replace 2 and 1/2 with 3 and 1/3 – so see n^{1/3} means take the cube root – and so on.

n^{1/x} means take the xth root of n.

Just before we get back to the question given, lets just complete the patterns by considering what x^{3/2} means. I have seen some exam questions that do pose questions like this.

I suggest you split the 3/2 into 1/2 x 3 or, n^{3/2} = (n^{1/2})^{3}

so 4^{3/2} = (sqrt(4)^{3}) = 2^{3} = 8. It is generally easier to take the ‘root part first. In a non calculator paper you’ll only be asked about roots you know.

Let’s get back, at last, to the original question.

64^{-1/2}– Take the 1/2 part first, that means square root, and the square root of 64 is 8. The – part means take the reciprocal 1/8

What Carol did was take 1/2 of 64, not the square root of 64, so your answer should include a sentence. ‘Carol didn’t know that a power of 1/2 means square root, not multiply by 1/2’ – then give the correct answer of 1/8.

I’ve made this into 2 blogs posts with lots of background but if you can remember the rules given in these posts, this question need not take you long in an exam,