Evaluate the fifth root of two to
the power of five.
In this question, we have both a
power or exponent of five and a root of five.
We can evaluate the part within the
root sign first, two to the power of five. Two to the power five is different
to two times five, which would give us 10. When we’re working out two to the
power of five, we have five twos written down all multiplied together. The first two times two would give
us four multiplied by another two gives eight by another two gives 16 and, finally,
by the fifth two gives us 32. So, two to the power of five is
32. We can put this value back into the
original problem. So, now, we need to work out the
fifth root of 32.
If we say that this fifth root is
equal to a value 𝑎, then we’re really asking, “what value of 𝑎 could we write down
five times and multiply to give 32?” But in fact, we already know the
answer to this problem, and it’s two. The fifth root of 32 is two. We could see this from the original
problem as well. When we take the fifth power of a
number and then we take the fifth root of it, these two are inverse operations. We’ve effectively undone the
results of taking the fifth power. Either way, we can demonstrate that
the fifth root of two to the power of five is two.