https://file.aiquickdraw.com/mathbot/file/1727623293756eu2ndiw9.jpg

I have reviewed the image. Let's begin solving the problems step by step:

### Nos. 31-35: Evaluate each expression or state if it is not a real number.

**31) \( \sqrt{-16} \)**  
The square root of a negative number is not a real number. It is an imaginary number.  
\[ \sqrt{-16} = 4i \]

**32) \( (-32)^{\frac{1}{3}} \)**  
Taking the cube root of a negative number is a real number. The cube root of -32 is -2.  
\[ (-32)^{\frac{1}{3}} = -2 \]

**33) \( \sqrt[3]{0} \)**  
The cube root of 0 is 0.  
\[ \sqrt[3]{0} = 0 \]

**34) \( \sqrt{-1} \)**  
This is the definition of the imaginary unit \(i\).  
\[ \sqrt{-1} = i \]

**35) \( (27)^{\frac{2}{3}} \)**  
We can first take the cube root of 27 and then square it.  
\[ (27)^{\frac{2}{3}} = \left( \sqrt[3]{27} \right)^2 = 3^2 = 9 \]

### Nos. 36-40: Simplify the radical expressions.

**36) \( \sqrt{2} - 5\sqrt{2} \)**  
This is a subtraction of like terms:  
\[ \sqrt{2} - 5\sqrt{2} = -4\sqrt{2} \]

**37) \( \frac{1}{16x^3y} \)**  
Since this is a fraction with no simplification possible, the expression remains:  
\[ \frac{1}{16x^3y} \]

**38) \( 3\sqrt{5} - 2\sqrt{5} + 7\sqrt{5} \)**  
Again, combine like terms:  
\[ 3\sqrt{5} - 2\sqrt{5} + 7\sqrt{5} = 8\sqrt{5} \]

**39) \( \sqrt{-81x^6y^8} \)**  
Take the square root of each factor:  
\[ \sqrt{-81x^6y^8} = \sqrt{-81} \times \sqrt{x^6} \times \sqrt{y^8} = 9i \times x^3 \times y^4 = 9ix^3y^4 \]

**40) \( \sqrt{-32x^{10}y^8} \)**  
Similar to the previous example:  
\[ \sqrt{-32x^{10}y^8} = \sqrt{-32} \times \sqrt{x^{10}} \times \sqrt{y^8} = 4i \times x^5 \times y^4 = 4ix^5y^4 \]

### Nos. 41-45: Rationalize the denominators.

**41) \( \frac{1}{\sqrt{3}} \)**  
Multiply both the numerator and the denominator by \( \sqrt{3} \):  
\[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]

**42) \( \frac{2}{1+\sqrt{2}} \)**  
Multiply both the numerator and denominator by the conjugate \( 1-\sqrt{2} \):  
\[ \frac{2}{1+\sqrt{2}} \times \frac{1-\sqrt{2}}{1-\sqrt{2}} = \frac{2(1-\sqrt{2})}{(1+\sqrt{2})(1-\sqrt{2})} = \frac{2(1-\sqrt{2})}{1-2} = \frac{2(1-\sqrt{2})}{-1} = -2 + 2\sqrt{2} \]

**43) \( \frac{2}{3\sqrt{11}} \)**  
Multiply by \( \sqrt{11} \) to rationalize the denominator:  
\[ \frac{2}{3\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{2\sqrt{11}}{3 \times 11} = \frac{2\sqrt{11}}{33} \]

**44) \( \frac{1+\sqrt{2}}{1-\sqrt{2}} \)**  
Multiply by the conjugate \( 1+\sqrt{2} \):  
\[ \frac{1+\sqrt{2}}{1-\sqrt{2}} \times \frac{1+\sqrt{2}}{1+\sqrt{2}} = \frac{(1+\sqrt{2})^2}{(1-\sqrt{2})(1+\sqrt{2})} = \frac{1 + 2\sqrt{2} + 2}{1 - 2} = \frac{3 + 2\sqrt{2}}{-1} = -3 - 2\sqrt{2} \]

**45) \( \frac{3-\sqrt{5}}{3+\sqrt{5}} \)**  
Multiply by the conjugate \( 3-\sqrt{5} \):  
\[ \frac{3-\sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}} = \frac{(3-\sqrt{5})^2}{(3+\sqrt{5})(3-\sqrt{5})} = \frac{9 - 6\sqrt{5} + 5}{9 - 5} = \frac{14 - 6\sqrt{5}}{4} = \frac{7}{2} - \frac{3\sqrt{5}}{2} \]

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Would you like any further clarification on these steps? Let me know if you have additional questions!

### Related Questions:
1. How do you simplify higher-order radicals like \( \sqrt[4]{x} \)?
2. What are the key properties of radicals in terms of combining and simplifying them?
3. How can you rationalize a denominator when there are binomials?
4. What are the differences between real numbers and complex numbers when evaluating radicals?
5. Can you describe the process of solving radical equations?

**Tip**: Always remember to simplify radicals as much as possible by factoring perfect squares (or other higher powers) out of the radicand.

Evaluate the expressions and simplify the radical expressions as stated in the problems from 31 to 45.

Step-by-step solutions for simplifying radicals, evaluating cube roots, handling imaginary numbers, and rationalizing denominators for problems 31-45. Learn how to solve square roots of negative numbers, combine like radical terms, and rationalize complex fractions. Suitable for students in Grades 10-12.

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