<h2>Steps:</h2>
So before we jump into these problems, we must keep this particular rule in mind:
- Product Rule of Radicals: √ab = √a × √b. Additionally, remember that √a × √a = a
<h3>9.</h3>
So remember that the perimeter is the sum of all the sides. In this case:

So firstly, using the product rule of radicals we need to simplify these radicals as such:

Now, combine like terms (since they all have the same like term √3, they can all be added up):

<u>Your final answer is 17√3 in.</u>
<h3>10a.</h3>
For this, we will be using the pythagorean theorem, which is
, where a and b are the legs of the right triangle and c is the hypotenuse of the triangle. In this case, √18 and √32 are our legs and we need to find the hypotenuse. Set up our equation as such:

From here we can solve for the hypotenuse. Firstly, solve the exponents (remember that square roots and squared power cancel each other out):

Next, add up the left side:

Lastly, square root both sides of the equation:

<u>The hypotenuse is √50 cm.</u>
<h3>10b.</h3>
Now, the process is similar to that of 9 so I will just show the steps to the final answer.

<u>The perimeter is 12√2 in.</u>
<h3>11.</h3>
Now, the process is still similar to question 9 but remember that this time we are working with <em>cube roots</em>.
![P=3\sqrt[3]{24} +\sqrt[3]{27}+2\sqrt[3]{81}\\\\3\sqrt[3]{24}=3*\sqrt[3]{8}*\sqrt[3]{3}=3*2*\sqrt[3]{3}=6\sqrt[3]{3}\\\sqrt[3]{27}=3\\2\sqrt[3]{81}=2*\sqrt[3]{27}*\sqrt[3]{3}=2*3*\sqrt[3]{3}=6\sqrt[3]{3}\\\\P=6\sqrt[3]{3}+3+6\sqrt[3]{3}\\P=3+12\sqrt[3]{3}](https://tex.z-dn.net/?f=P%3D3%5Csqrt%5B3%5D%7B24%7D%20%2B%5Csqrt%5B3%5D%7B27%7D%2B2%5Csqrt%5B3%5D%7B81%7D%5C%5C%5C%5C3%5Csqrt%5B3%5D%7B24%7D%3D3%2A%5Csqrt%5B3%5D%7B8%7D%2A%5Csqrt%5B3%5D%7B3%7D%3D3%2A2%2A%5Csqrt%5B3%5D%7B3%7D%3D6%5Csqrt%5B3%5D%7B3%7D%5C%5C%5Csqrt%5B3%5D%7B27%7D%3D3%5C%5C2%5Csqrt%5B3%5D%7B81%7D%3D2%2A%5Csqrt%5B3%5D%7B27%7D%2A%5Csqrt%5B3%5D%7B3%7D%3D2%2A3%2A%5Csqrt%5B3%5D%7B3%7D%3D6%5Csqrt%5B3%5D%7B3%7D%5C%5C%5C%5CP%3D6%5Csqrt%5B3%5D%7B3%7D%2B3%2B6%5Csqrt%5B3%5D%7B3%7D%5C%5CP%3D3%2B12%5Csqrt%5B3%5D%7B3%7D)
Note that when adding the numbers together, 3 isn't a like term to the other 2 terms because it doesn't have ∛3 multiplied with it.
<u>The perimeter is 3 + 12∛3 in.</u>