Radical Equivalent to Rational Exponents 2

u16_l1_t3_we2 Radical Equivalent to Rational Exponents 2

Rewrite the radical expression

using rational exponents and simplify.

So here we have the fourth root of 5 a to the fourth

b to the twelfth power.

The key thing to realize here is the fourth root of something

is same thing as something to the one fourth power.

Or in particular, or in general, the nth root something

is the same thing as that something to the 1 over n power.

So we can just apply that over here:

the fourth root of all of this

is equal to 5 a to the 4th, b to the 12th power,

all of that to the one fourth power.

And then we also know if we take the product of things,

and then raise them to some exponent,

that's the same thing as raising each of the terms in the product

to the exponent first,

or each of the things that we're taking the product of

to that exponent, and then multiplying.

So let's do that.

So 5, so this is the same thing as

5 to the 1/4 power

times a to the fourth to the 1/4 power

times b to the 12th to the 1/4 power.

Now 5 to the 1/4,

I don't know what that is,

so I'll just keep that as the cube root,

well, we could leave it as 5 to the 1/4,

and that's not not simplified.

or we can just rewrite it again as the fourth root of 5.

a to the fourth to the 1/4 power:

if your raise something to a power and then another power,

and raise that to another power,

that's equivalent of raising a to the four times 1/4 power.

So let me just write that down.

This is, so times a to four times 1/4 power.

And then finally, this right over here,

using the same exact exponent property,

this is b to the 12th times 1/4 power.

So all of this simplifies to,

and I'll change the order here,

so you have the fourth root of 5,

and then you have a to the fourth times 1/4 power

so that's just, this simplifies to a to the first power

which is really just the same thing as a.

So that's just a.

And then we have b to the 12 times 1/4 power.

Well, 12 times 1/4 is just three.

So that's b to the third power.

So it's a, b to the third power times the fourth root of 5.