We want to simplify the given radical
expression and we'll show how to simplify this two ways.
We'll first use rational exponents.
And then we'll simplify it in radical form.
To write this expression using rational exponents,
we'll start with the innermost radical,
which is the square root of 64.
Because we have a square root, the index is two
and the exponent around 64 would be one.
And therefore in rational exponent form,
this would be equal to, we'd start with 64
raised to the power of one half.
Remember when going from radical form to rational
exponent form, the index gives us the denominator
of the rational exponent.
And the exponent gives us the numerator
of the rational exponent.
So 64 to the one half power is the same as the
square root of 64 and then we're taking the cubit of this,
which would be equivalent to raising this to the
one-third power.
And we're taking the fourth root of this,
which would be equivalent to raising this to
the one-fourth power.
And all this is raised to the eighth power,
so we have all of this raised to the eighth.
Then I simplify this because we have power to the powers,
we multiply the exponents, given by this exponent rule here.
We have a raised to the power of m,
raised to the power of n, equals a raised to the
power of m times n.
So we would have 64 raised to the power of one half
times one third times one fourth,
times eight, which I'll write as eight over one.
Now before multiplying our exponents, let's simplify.
Notice the denominator we have two times four
which equals eight.
We also have an eight in the numerator.
And eight divided by eight simplifies to one.
So this simplifies to 64 raised to the power of one-third.
And now to simplify 64 to the one-third power,
we'll write this back in radical form.
Notice how the denominator of the rational exponent is three
so we have a cube root.
And we have 64 raised to the first power,
so we can leave off the exponent.
64 is a perfect cube and therefore this
does simplify perfectly.
64 is equal to four to the third.
And therefore the cubit of four to the third is
equal to one factor of four.
So this entire expression does simplify to four.
But then I'll show how to simplify this a different way
leaving it in radical form as long as possible.
So again we now know this does equal four.
But we're going to show the simplification
using a different method.
So if we simplify this by starting with the innermost
radical which is the square root of 64,
because 64 is a perfect square, this simplifies perfectly.
The square root of 64 equals the square root of eight
squared which equals eight.
So if we simplify the square root of 64 to eight,
this would leave us with the cube root
of eight and then we're going to take the fourth root
of this and all this is still raised to the eighth power.
Now looking at the innermost radical again,
now we have the cubit of eight,
which again does simplify perfectly because eight
is a perfect cube, eight is equal to two cubed,
or two times two times two.
And therefore the cubit of eight would be one factor of two.
So this would simplify to the fourth root of two
all raised to the eighth.
And now at this point, we cannot simplify the fourth
root of two, so we rewrite this using rational exponents.
So two is the same as two to the first.
So inside the parentheses, we would have two raised
to the one-fourth all raised to the eighth.
And again here we have powers to powers
so we'd multiply the exponents.
One-fourth times eight is equal to eight fourths, or two.
So this simplifies to two squared, which equals four.
So we get the same result using a different method.
I hope you found this helpful.