Question

How do you simplify `sqrt(z^3)sqrt(z^7)`?

Answer 1

We use the properties of powers to find `sqrt(z^3) * sqrt(z^7) = z^5`

Explanation:

We need to write the powers of the factors in a single number so that we can add them. We start by noting that a square root is just a power of `1/2`, i.e.:

`sqrt(z^3) * sqrt(z^7) = (z^3)^(1/2) * (z^7)^(1/2)`

Now we use the property that raising a power to another power multiplies the exponents:

`(z^3)^(1/2) * (z^7)^(1/2) = z^(3/2) * z^(7/2)`

Then we use the property that multiplying powers with the same base adds the exponents:

`z^(3/2) * z^(7/2) = z^(3/2+7/2) = z^(10/2) = z^5`