Simplify 9 square root of 2( square root of 15+ square root of 14)

Solution:

Step 1:

Utilize the distributive property to expand the expression: $9\sqrt{2}(\sqrt{15}+\sqrt{14})$.

Step 2:

Perform multiplication for $9\sqrt{2}\cdot \sqrt{15}$.

Step 2.1:

Apply the radical multiplication rule: $9\sqrt{2\cdot 15}+9\sqrt{2}\cdot \sqrt{14}$.

Step 2.2:

Calculate the product of $15$ and $2$: $9\sqrt{30}+9\sqrt{2}\cdot \sqrt{14}$.

Step 3:

Perform multiplication for $9\sqrt{2}\cdot \sqrt{14}$.

Step 3.1:

Apply the radical multiplication rule: $9\sqrt{30}+9\sqrt{14\cdot 2}$.

Step 3.2:

Calculate the product of $14$ and $2$: $9\sqrt{30}+9\sqrt{28}$.

Step 4:

Simplify the radical expressions.

Step 4.1:

Express $28$ as a product of its prime factors: $2^2\cdot 7$.

Step 4.1.1:

Extract the square root of $4$ from $\sqrt{28}$: $9\sqrt{30}+9\sqrt{4\cdot 7}$.

Step 4.1.2:

Represent $4$ as $2^2$: $9\sqrt{30}+9\sqrt{2^2\cdot 7}$.

Step 4.2:

Extract terms from under the radical sign: $9\sqrt{30}+9(2\sqrt{7})$.

Step 4.3:

Multiply $2$ by $9$: $9\sqrt{30}+18\sqrt{7}$.

Step 5:

Present the final result in various forms.

Exact Form: $9\sqrt{30}+18\sqrt{7}$

Decimal Form: $96.91855377\dots$

Knowledge Notes:

1. Distributive Property: This property states that for any real numbers $a$, $b$, and $c$, the equation $a(b+c)=ab+ac$ holds true. It allows us to multiply a single term by each term within a parenthesis.

2. Product Rule for Radicals: The product rule for radicals states that $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$, provided $a$ and $b$ are nonnegative real numbers. This rule is used to combine or separate products under a single radical.

3. Simplifying Radicals: To simplify a radical, one should look for factors that are perfect squares (or higher powers if dealing with cube roots, etc.) and extract them from under the radical sign. For example, $\sqrt{4\cdot 7}$ can be simplified to $2\sqrt{7}$ because $4$ is a perfect square.

4. Prime Factorization: This is the process of breaking down a composite number into its prime factors. For example, $28$ can be factored into $2^2\cdot 7$. This is useful in simplifying radicals because it helps to identify and extract perfect square factors.

5. Exact vs. Decimal Form: The exact form of an expression includes radicals and provides an exact value. The decimal form is an approximation of the exact value, often rounded to a certain number of decimal places. It is important to know when to use each form, as the exact form is often preferred in mathematical proofs and calculations to avoid rounding errors.