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I see the image you've uploaded includes several sections on simplifying radicals and working with variables under square roots. Let's go through each section step by step.

### Section D: Simplifying Radicals (Square Roots)

1. \( \sqrt{121} = \boxed{11} \)
2. \( \sqrt{36} + \sqrt{25} = 6 + 5 = \boxed{11} \)
3. \( \sqrt{100} \times \sqrt{49} = 10 \times 7 = \boxed{70} \)
4. \( \sqrt{64} + \sqrt{9} \times 3 = 8 + 3 \times 3 = 8 + 9 = \boxed{17} \)
5. \( \sqrt{144} - 2 \times \sqrt{25} + \sqrt{16} = 12 - 2 \times 5 + 4 = 12 - 10 + 4 = \boxed{6} \)
6. \( 4 \times \sqrt{81} - 5 \times \sqrt{4} + \sqrt{64} = 4 \times 9 - 5 \times 2 + 8 = 36 - 10 + 8 = \boxed{34} \)

### Section E: Simplifying Radicals with Variables (Including Odd Powers)

1. \( \sqrt{y^2} = \boxed{y} \)
2. \( \sqrt{9p^2} = 3p \)
3. \( 2 \times \sqrt[3]{16q^3} = 2 \times 2q = \boxed{4q} \)
4. \( \sqrt{36r^2} + 3 \times \sqrt[3]{4s^3} = 6r + 3 \times 2s = \boxed{6r + 6s} \)
5. \( \sqrt{49t^3} + \sqrt{64u^2} = 7t^{3/2} + 8u \)
6. \( \sqrt{25v^3} - \sqrt{9w^5} = 5v^{3/2} - 3w^{5/2} \)
7. \( \sqrt{81x^5} + 2 \times \sqrt[3]{16y^3} = 9x^{5/2} + 4y \)
8. \( \sqrt{121z^3} - \sqrt{64a^5} + \sqrt[3]{100b^3} = 11z^{3/2} - 8a^{5/2} + 10b \)

### Challenge Section (Optional)

1. \( |-7 + 4| + 5 \times \sqrt{49} - 9 = 3 + 5 \times 7 - 9 = 3 + 35 - 9 = \boxed{29} \)
2. Given \( x = 8 \), \( y = -3 \), and \( z = 25 \):
   \( \sqrt{x} + |y| - \sqrt{z} + 6 = \sqrt{8} + 3 - 5 + 6 \approx 2.83 + 3 - 5 + 6 = \boxed{6.83} \)
3. \( 4 \times \sqrt{36p^2} + |15 + 9| - 4 = 4 \times 6p + 24 - 4 = 24p + 20 = \boxed{24p + 20} \)

Let me know if you'd like any further details or explanations!

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Here are some related questions to deepen your understanding:

1. How do you simplify a square root when the radicand includes variables with exponents?
2. What is the process for evaluating radicals when combining multiple terms?
3. How do odd and even powers affect simplifying radicals with variables?
4. What are the rules for handling absolute value in an expression like in the challenge problem?
5. How can you estimate the value of non-perfect square roots when calculating them by hand?

**Tip:** When simplifying radicals, always check if the number or variable under the root can be factored into perfect squares, as this greatly simplifies the process!

Simplify the radicals and expressions in the worksheet, including sections on simplifying square roots and radicals with variables.

Explore how to simplify radicals, including those with variables and odd powers, through this detailed worksheet. Covers essential algebraic concepts like square roots, properties of radicals, and absolute values. Suitable for students in Grades 8-10.

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