Video Transcript
Complete the following: The sixth
root of 64 equals the cube root of blank.
In this question, on the right-hand
side, we have an unknown value which has the third or the cube root taken off
it. The cube root of this unknown value
is equal to the sixth root of 64. So, perhaps the best place to start
is to actually work out the sixth root of 64.
If we say that the sixth root of 64
would give us this value, which we can call 𝑥, it’s equivalent to writing 𝑥 six
times and multiplying. And that would give us a value of
64. The question is, what is 𝑥? If we were to try some different
values of 𝑥 and we started with one, well, one written six times and multiplied
would still give us a value of one, which isn’t the value of 64 that we were looking
for.
So, how about a value of 𝑥 equals
two? There’s a number of different ways
to work this out, but we need to be careful that we’re not multiplying six by
two. If we began by multiplying the
first two times two, that would give us an answer of four. And we could then multiplied it by
the next two, which would give us an answer of eight. We might then notice that we have
another two times two times two, which would also give us eight, and multiplying
eight times eight would give us 64.
That means that we have worked out
that the sixth root of 64 is two. And so, the left-hand side of our
equation should be equal to two, and so should the right-hand side. We should be careful when we’re
answering questions like this, as if we just wrote in the value of two on the
missing line, it would be incorrect.
Since this expression on the
right-hand side is equal to two, we need to think what value taken to the third root
would give us an answer of two. The inverse of finding the third
root is to find the third power, so we need to calculate two to the third power. And it’s eight since two times two
times two is eight. We can, therefore, give the missing
answer as eight since the sixth root of 64 is equal to two and the cube root of
eight is also equal to two.
