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
n0 = 1 and n1 = n.
With negative powers we still need to maintain the rule of adding powers.
2x x 2-x = 2x-x (and since x – x x = 0) = 20 = 1
Re-arrange this and we get 2-x = 1/2x
And there the first new rule – a negative power is 1/the positive power
32 = 9 so 3-2 = 1/9
To get the second rule we need to consider how powers can be combined.
(n2)3 = n6 – 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
(n2)1/2 = n1 = n
So what does raise to power of half mean if this involves getting from n2 to n? Taking the square root! We could replace 2 and 1/2 with 3 and 1/3 – so see n1/3 means take the cube root – and so on.
n1/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 x3/2 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, n3/2 = (n1/2)3
so 43/2 = (sqrt(4)3) = 23 = 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,