What is one-eighth of the seventh root of eight all to the power of 14? Well we can rewrite this expression slowly. One-eighth of something means one-eighth times it. So we’ve got an eighth of the seventh root of eight all to the power of 14.
Now I’m gonna express the seventh root of eight in a slightly different format. And that’s eight to the power of a seventh. Eight to the power of a seventh is the same as the seventh root of eight.
So that’s a general rule; we can express the 𝑏th root of something as 𝑥 to the power of one over 𝑏. But there’s another general rule; that 𝑥 to the power of 𝑎 all to the power of 𝑏 can be expressed as 𝑥 to the 𝑎 times 𝑏.
Now this means that I can rewrite eight to the power of a seventh all to the power of 14 as eight to the power of a seventh times 14. And I can re-express 14 as 14 over one to make it clear that a seventh times 14 or one over seven times 14 over one is the same as 14 over seven, which is two.
So we’ve ended up with an eighth times eight squared. Now remember eight squared means eight times eight.
And again I can express those as eight over one, so I’ve got one times eight times eight over eight times one times one. Now if I divide the top by eight, I get one; if I divide the bottom by eight, I get one. So I’ve got one times one times eight over one times one times one, which is just eight.
So my answer is eight.