Simplify the following: .

Rewriting things in fractional form is definitely the way to kick things off.

Next, we can use our power rule, which tells us to multiply when one exponent is raised to another.

(x²)^1/3(x⁵) = (x^2/3)(x⁵)

(x^2/3)(x⁵) = x^{2/3 + 5} = x^{2/3 + 15/3} = x^17/3

Simplify the following: .

Once again, making exponents fractional is a must.

Multiplying exponents will make our final step pretty simple. Note that we took the long way around in simplifying , which is really just 3 or 3¹.

(3²)^1/2(3⁴)^1/5 = (3¹)(3^4/5)

Finally, we can just add the exponents since the bases are the same.

(3¹)(3^4/5) = 3^{1 + 4/5} = 3^{5/5 + 4/5} = 3^9/5

Simplify the following: .

This problem is pretty similar to the previous example, except we're adding the two terms instead of multiplying 'em. Either way, it's important to take care of the exponent fractionalization first.

Next, we need to decide if we can simplify when the bases are the same but the exponents are different. Whenever we get stuck like this, we find it helpful to go back to a situation we know, like x² + x³. We wouldn't combine these terms, which means we have no grounds to combine the terms in our problem either. Done and done.

3¹ + 3^4/5 = 3 + 3^4/5

Simplify the following using only positive exponents: .

First, let's work to simplify inside that square root.

Next, we can make the seventh root into a fractional exponent. This is really nice because we can then treat each factor separately.

Now that we have fractional exponents, we can find the seventh root of 128 and simplify our exponents by multiplying.

128^1/7(a²)^1/7(b⁷)^1/7 = 2a^2/7b^7/7

We're actually just about done here. All we have to do is simplify our fractional exponents.

2a^2/7b^7/7 = 2a^2/7b