Simplify square root of 2( square root of 8+4 square root of 3)

Solution:

Step:1

Break down each term within the expression.

Step:1.1

Express $8$ as a product of $2^2$ and $2$.

Step:1.1.1

Extract $4$ as a common factor from $8$. $\sqrt{2} \left( \sqrt{4 \cdot 2} + 4 \sqrt{3} \right)$

Step:1.1.2

Represent $4$ as $2^2$. $\sqrt{2} \left( \sqrt{2^2 \cdot 2} + 4 \sqrt{3} \right)$

Step:1.2

Extract terms from under the square root. $\sqrt{2} \left( 2 \sqrt{2} + 4 \sqrt{3} \right)$

Step:2

Use the distributive law to expand. $\sqrt{2} \cdot 2 \sqrt{2} + \sqrt{2} \cdot 4 \sqrt{3}$

Step:3

Calculate $\sqrt{2} \cdot 2 \sqrt{2}$.

Step:3.1

Elevate $\sqrt{2}$ to the power of $1$. $2 \left( \left(\sqrt{2}\right)^1 \sqrt{2} \right) + \sqrt{2} \cdot 4 \sqrt{3}$

Step:3.2

Apply the exponent of $1$ to $\sqrt{2}$. $2 \left( \left(\sqrt{2}\right)^1 \left(\sqrt{2}\right)^1 \right) + \sqrt{2} \cdot 4 \sqrt{3}$

Step:3.3

Utilize the rule $a^{m} a^{n} = a^{m + n}$ to combine the powers. $2 \left(\sqrt{2}\right)^{1 + 1} + \sqrt{2} \cdot 4 \sqrt{3}$

Step:3.4

Sum the exponents $1$ and $1$. $2 \left(\sqrt{2}\right)^2 + \sqrt{2} \cdot 4 \sqrt{3}$

Step:4

Compute $\sqrt{2} \cdot 4 \sqrt{3}$.

Step:4.1

Merge using the radical product rule. $2 \left(\sqrt{2}\right)^2 + 4 \sqrt{2 \cdot 3}$

Step:4.2

Multiply $2$ and $3$. $2 \left(\sqrt{2}\right)^2 + 4 \sqrt{6}$

Step:5

Simplify each component.

Step:5.1

Convert $\left(\sqrt{2}\right)^2$ back to $2$.

Step:5.1.1

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$. $2 \left(2^{\frac{1}{2}}\right)^2 + 4 \sqrt{6}$

Step:5.1.2

Apply the rule $\left(a^{m}\right)^{n} = a^{m \cdot n}$. $2 \cdot 2^{\frac{1}{2} \cdot 2} + 4 \sqrt{6}$

Step:5.1.3

Multiply $\frac{1}{2}$ by $2$. $2 \cdot 2^{\frac{2}{2}} + 4 \sqrt{6}$

Step:5.1.4

Simplify the fraction.

Step:5.1.4.1

Reduce the common factors. $2 \cdot 2^{\frac{\cancel{2}}{\cancel{2}}} + 4 \sqrt{6}$

Step:5.1.4.2

Express the simplified power. $2 \cdot 2^1 + 4 \sqrt{6}$

Step:5.1.5

Compute the power of $2$. $2 \cdot 2 + 4 \sqrt{6}$

Step:5.2

Multiply $2$ by $2$. $4 + 4 \sqrt{6}$

Step:6

Present the final result in various forms.

Exact Form: $4 + 4 \sqrt{6}$ Decimal Form: $13.79795897 \ldots$

Knowledge Notes:

The problem involves simplifying a nested radical expression. The key knowledge points used in the solution include:

1. Radicals: Understanding how to simplify square roots and the properties of radicals is essential. For instance, $\sqrt{a^2} = a$ when $a$ is non-negative.

2. Exponent Rules: Familiarity with exponent rules, such as $a^{m} a^{n} = a^{m + n}$ and $\left(a^{m}\right)^{n} = a^{m \cdot n}$, is crucial for simplifying expressions involving powers.

3. Distributive Property: This property allows us to multiply a single term by each term within a parenthesis, which is used in Step 2.

4. Combining Like Terms: When simplifying expressions, like terms can be combined to form a simpler expression.

5. Product Rule for Radicals: This rule states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, which is used to combine square roots in Step 4.

6. Simplifying Expressions: The process of reducing expressions to their simplest form by performing arithmetic operations and applying algebraic rules.