Problem 280 - Elementary Algebra

Question

In the following exercises, simplify. $$ (\sqrt{3}+\sqrt{10})(\sqrt{3}+2 \sqrt{10}) $$

Step-by-Step Solution

Answer

23 + 3\sqrt{30}

Step 1: Expand the Expression

Use the distributive property (also known as FOIL method for binomials) to expand the expression \((\begin{align*}(\sqrt{3} + \sqrt{10})(\sqrt{3} + 2 \sqrt{10}))\end{align*}:\)First, multiply each term in the first binomial by each term in the second binomial.\(\begin{align*}=& (\sqrt{3} \cdot \sqrt{3}) + (\sqrt{3} \cdot 2 \sqrt{10}) + (\sqrt{10} \cdot \sqrt{3}) + (\sqrt{10} \cdot 2 \sqrt{10})\end{align*}\)

Step 2: Simplify Each Term

Simplify each term from the expansion. \(\begin{align*}& \sqrt{3} \cdot \sqrt{3} = 3 \,\text{(since} \sqrt{a} \cdot \sqrt{a} = a\text{)}\& \sqrt{3} \cdot 2 \sqrt{10} = 2 \sqrt{30} \& \sqrt{10} \cdot \sqrt{3} = \sqrt{30} \& \sqrt{10} \cdot 2 \sqrt{10} = 2 \cdot 10 = 20\end{align*}\)Thus, we have: \(\begin{align*}3 + 2 \sqrt{30} + \sqrt{30} + 20\end{align*}\)

Step 3: Combine Like Terms

Combine the like terms to further simplify the expression. \(\begin{align*}& 3 + 20 = 23\& 2 \sqrt{30} + \sqrt{30} = 3 \sqrt{30}\end{align*}\)Therefore, the simplified expression is: \(\begin{align*}23 + 3 \sqrt{30}\end{align*}\)

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra. It allows us to multiply a sum by distributing the multiplication over each term within parentheses. In this exercise, we start by applying the distributive property, also known as the FOIL method when dealing with binomials. This involves multiplying each term in the first binomial by every term in the second binomial.

For the given expression \[ (\begin{align*} (\sqrt{3} + \sqrt{10})(\sqrt{3} + 2\sqrt{10})\begin{align*}) \], we use the distributive property to expand: \[(\sqrt{3} \cdot \sqrt{3}) + (\sqrt{3} \cdot 2\sqrt{10}) + (\sqrt{10} \cdot \sqrt{3}) + (\sqrt{10} \cdot 2\sqrt{10}) \]

This step ensures each term is multiplied properly, preparing us for further simplification.

Radical Expressions

Radical expressions include roots, such as square roots (e.g., \sqrt{3} and \sqrt{10}). These roots can often be simplified before or as part of further calculations.

When simplifying each term in the expanded expression, we use the property \[ \sqrt{a} \cdot \sqrt{a} = a \]. This allows us to simplify:

• \sqrt{3} \cdot \sqrt{3} = 3 \text {(as \sqrt{3} times \sqrt{3} gives 3)}
• \sqrt{3} \cdot 2 \sqrt{10} = 2 \sqrt{30}
• \sqrt{10} \cdot \sqrt{3} = \sqrt{30}
• \sqrt{10} \cdot 2\sqrt{10} = 2 \cdot 10 = 20

With these simplifications, we rewrite the expression as: \[ 3 + 2 \sqrt{30} + \sqrt{30} + 20 \]

Combining Like Terms

Combining like terms is a key step to further simplify an algebraic expression. Like terms are terms that contain the same variables raised to the same power. In the context of our exercise, we look at numerical coefficients and radical parts.

When we have: \[ 3 + 2\sqrt{30} + \sqrt{30} + 20 \] we combine like terms as follows:

• Combine the constants: \ 3 + 20 = 23
• \sqrt{30} terms: \ 2\sqrt{30} + \sqrt{30} = 3 \sqrt{30}

The simplified expression becomes: \[ 23 + 3\sqrt{30} \]
This final expression shows a single constant term and a single term involving the radical, making our expression much simpler and easier to understand.