Simplify square root of 6( square root of 3+5 square root of 2)

Solution:

Step 1:

Utilize the distributive property to expand: $\sqrt{6}(\sqrt{3} + 5\sqrt{2})$ becomes $\sqrt{6}\sqrt{3} + \sqrt{6}(5\sqrt{2})$.

Step 2:

Apply the product rule for radicals to combine terms: $\sqrt{6 \cdot 3} + \sqrt{6}(5\sqrt{2})$.

Step 3:

Perform the multiplication for the second term: $\sqrt{6}(5\sqrt{2})$.

Step 3.1:

Use the product rule for radicals to combine the second term: $\sqrt{6 \cdot 3} + 5\sqrt{6 \cdot 2}$.

Step 3.2:

Calculate the product inside the radical for the second term: $\sqrt{6 \cdot 3} + 5\sqrt{12}$.

Step 4:

Simplify the radical expressions.

Step 4.1:

Multiply inside the first radical: $\sqrt{18} + 5\sqrt{12}$.

Step 4.2:

Express 18 as a product of its prime factors: $18 = 3^2 \cdot 2$.

Step 4.2.1:

Extract the square factor from under the radical: $\sqrt{9 \cdot 2} + 5\sqrt{12}$.

Step 4.2.2:

Represent 9 as $3^2$: $\sqrt{3^2 \cdot 2} + 5\sqrt{12}$.

Step 4.3:

Extract square factors from under the radicals: $3\sqrt{2} + 5\sqrt{12}$.

Step 4.4:

Express 12 as a product of its prime factors: $12 = 2^2 \cdot 3$.

Step 4.4.1:

Extract the square factor from under the radical: $3\sqrt{2} + 5\sqrt{4 \cdot 3}$.

Step 4.4.2:

Represent 4 as $2^2$: $3\sqrt{2} + 5\sqrt{2^2 \cdot 3}$.

Step 4.5:

Extract square factors from under the radicals: $3\sqrt{2} + 5(2\sqrt{3})$.

Step 4.6:

Perform the multiplication for the second term: $3\sqrt{2} + 10\sqrt{3}$.

Step 5:

Present the final result in its various forms.

Exact Form: $3\sqrt{2} + 10\sqrt{3}$

Decimal Form: $21.56314876\ldots$

Knowledge Notes:

To solve the given problem, we use several mathematical properties and rules:

1. Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. For instance, $a(b + c) = ab + ac$.

2. Product Rule for Radicals: This rule states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, allowing us to combine or separate products under a single radical.

3. Simplifying Radicals: When a radical contains a perfect square, we can simplify it by taking the square root of the perfect square out of the radical. For example, $\sqrt{9 \cdot a} = 3\sqrt{a}$ because $9$ is a perfect square.

4. Prime Factorization: This involves expressing a number as the product of its prime factors, which can be useful when simplifying radicals. For example, $18$ can be written as $3^2 \cdot 2$.

5. Extracting Square Factors: When a radical contains a term that is a perfect square, we can "pull out" the square root of that term. For instance, $\sqrt{2^2 \cdot a} = 2\sqrt{a}$.

By applying these rules and properties step by step, we can simplify the given expression to its simplest form.