https://r2.erweima.ai/mathbot/image/17267555959792ase6334.jpg

Let's solve the problems from the image you uploaded. The instructions are to apply mathematical properties when possible. Here’s the solution for each part:

### a) \((\sqrt{8} + \sqrt{2})^2\)
Apply the binomial expansion formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[
\left( \sqrt{8} + \sqrt{2} \right)^2 = \left( \sqrt{8} \right)^2 + 2\left( \sqrt{8} \right)\left( \sqrt{2} \right) + \left( \sqrt{2} \right)^2
\]
First, simplify the square roots:
\[
\left( \sqrt{8} \right)^2 = 8, \quad \left( \sqrt{2} \right)^2 = 2, \quad 2 \left( \sqrt{8} \cdot \sqrt{2} \right) = 2 \sqrt{16} = 2 \times 4 = 8
\]
Now add them together:
\[
8 + 8 + 2 = 18
\]
So, the result is:
\[
\boxed{18}
\]

### b) \(\sqrt{50}\)
Simplify \(\sqrt{50}\):
\[
\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}
\]
So, the result is:
\[
\boxed{5\sqrt{2}}
\]

### c) \(\sqrt{100} - \sqrt{36}\)
Evaluate each square root:
\[
\sqrt{100} = 10, \quad \sqrt{36} = 6
\]
Now subtract them:
\[
10 - 6 = 4
\]
So, the result is:
\[
\boxed{4}
\]

### d) \((a^3 \cdot a)^2 : (a^4 \cdot a)\)
Simplify using the properties of exponents:
\[
(a^3 \cdot a)^2 = a^{(3+1) \cdot 2} = a^8, \quad a^4 \cdot a = a^{4+1} = a^5
\]
Now, divide:
\[
\frac{a^8}{a^5} = a^{8-5} = a^3
\]
So, the result is:
\[
\boxed{a^3}
\]

### e) \(\frac{4 \cdot \sqrt{81}}{64}\)
Simplify:
\[
\sqrt{81} = 9
\]
Now calculate:
\[
\frac{4 \cdot 9}{64} = \frac{36}{64} = \frac{9}{16}
\]
So, the result is:
\[
\boxed{\frac{9}{16}}
\]

### f) \(3 \cdot \sqrt{12}\)
Simplify \(\sqrt{12}\):
\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
\]
Now multiply:
\[
3 \cdot 2\sqrt{3} = 6\sqrt{3}
\]
So, the result is:
\[
\boxed{6\sqrt{3}}
\]

### g) \(\sqrt{36 + 64}\)
First, add the numbers inside the square root:
\[
36 + 64 = 100
\]
Now take the square root:
\[
\sqrt{100} = 10
\]
So, the result is:
\[
\boxed{10}
\]

---

Would you like a detailed explanation of any specific step? Here are some related questions:

1. How does the binomial expansion formula work in simplifying expressions?
2. How can you further simplify expressions involving square roots?
3. What are the rules for multiplying and dividing exponents?
4. How can we simplify square roots when they involve products of numbers?
5. What are some common properties of square roots that can be used in simplifying equations?

**Tip:** When simplifying square roots, always look for perfect square factors to break them down easily!

Resuelve aplicando propiedades cuando sea posible. a) (√8 + √2)^2, b) √50, c) √100 - √36, d) (a^3.a)^2 : (a^4.a), e) 4√81 ÷ 64, f) 3√12, g) √36 + 64.

Learn how to apply mathematical properties to simplify square roots, binomials, and exponents. This problem covers square root simplification, binomial expansion, and exponent properties for Grades 9-11 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation