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

Jul 12, 2016

We use the properties of powers to find $\sqrt{{z}^{3}} \cdot \sqrt{{z}^{7}} = {z}^{5}$

#### 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 $\frac{1}{2}$, i.e.:

$\sqrt{{z}^{3}} \cdot \sqrt{{z}^{7}} = {\left({z}^{3}\right)}^{\frac{1}{2}} \cdot {\left({z}^{7}\right)}^{\frac{1}{2}}$

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

${\left({z}^{3}\right)}^{\frac{1}{2}} \cdot {\left({z}^{7}\right)}^{\frac{1}{2}} = {z}^{\frac{3}{2}} \cdot {z}^{\frac{7}{2}}$

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

${z}^{\frac{3}{2}} \cdot {z}^{\frac{7}{2}} = {z}^{\frac{3}{2} + \frac{7}{2}} = {z}^{\frac{10}{2}} = {z}^{5}$