Problem 33 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{3}(5 \sqrt{2}+\sqrt{3}) $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(5\sqrt{6} + 3\).

Step 1: Distribute \(\sqrt{3}\)

Use the distributive property to multiply \(\sqrt{3}\) to both terms. \(\sqrt{3} \cdot 5\sqrt{2} = 5\sqrt{6}\) and \(\sqrt{3} \cdot \sqrt{3} = 3\). This results in \(5\sqrt{6}+3\).

Step 2: Write the final Simplified Expression

After performing the previous step, the expression is already simplified. Hence, the final simplified expression is \(5\sqrt{6} + 3\).

Key Concepts

Distributive Property

Understanding the distributive property is essential when working with algebraic expressions. It's the rule that enables us to multiply a single term by each term within a parenthesis in an expression. Formally, it is expressed as: \( a(b + c) = ab + ac \). When dealing with radical expressions like \(\sqrt{3}(5 \sqrt{2}+\sqrt{3})\), applying the distributive property means multiplying \(\sqrt{3}\) with each term inside the parenthesis individually.

To aid visualization:

• First, \(\sqrt{3}\times 5\sqrt{2} = 5\sqrt{6}\) because you multiply the numbers outside the radicals and keep the product under a single radical.
• Next, \(\sqrt{3}\times \sqrt{3} = 3\), since the square root of a number times itself gives the number.

As seen, each term inside the parenthesis has been multiplied by \(\sqrt{3}\) to distribute the value evenly across the expression.

Radical Simplification

Radical simplification is the process of making a radical expression less complex. This can involve combining like terms, rationalizing the denominator, or finding square factors of the number under the radical sign that can be taken out as whole numbers.

For simplifying the expression \(\sqrt{3}(5 \sqrt{2}+\sqrt{3})\), we used the property that \(\sqrt{a}\times\sqrt{b} = \sqrt{ab}\) to simplify the products. After distributing \(\sqrt{3}\), we got \(5\sqrt{6}+3\). There are no like terms to combine and no further square factors under the radicals. Thus, this is the simplest form of the expression.

Square Roots

Square roots are a fundamental concept in mathematics that deal with finding a number which, when multiplied by itself, gives the original number. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

When simplifying expressions with square roots, we often look for perfect squares that can be easily rooted. In the given exercise, we encounter \(\sqrt{3} \times \sqrt{3}\), which simplifies to \(3\) as \(3\) is the square root of \(9\), making the process straightforward. The collective understanding of square roots is key for radical simplification and employing the distributive property in radical expressions.