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

1 Answer
Jul 12, 2016

Answer:

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#