Question

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

Short answer

The simplified form of the expression \(\sqrt{3}(5\sqrt{2}+\sqrt{3})\) is \(5\sqrt{6}+3\).

Step-by-step solution

Distribute \(\sqrt{3}\) into the parentheses

Distribute \(\sqrt{3}\) to both \(5\sqrt{2}\) and \(\sqrt{3}\) inside the parenthesis. Remember that when you multiply a square root times itself, it simplifies to the number itself.

Simplify the result

The result of distributing \(\sqrt{3}\) to both \(5\sqrt{2}\) and \(\sqrt{3}\) will then give you \(5\sqrt{6}+3\).

Key concepts

Radical Expressions

Understanding radical expressions is essential when dealing with square roots and higher-order roots. A radical expression is an expression that contains a square root, cube root, or higher root. The general form is \( \sqrt[n]{x} \) where \( n \) is the root degree and \( x \) is the radicand (the number under the root symbol).

Simplifying radical expressions often involves merging the roots of products, which employs the mathematical principle that \( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \) if both \( a \) and \( b \) are non-negative. This property streamlines complex radical expressions into simpler, more manageable forms.

In the given exercise, we start with \( \sqrt{3}(5 \sqrt{2}+\sqrt{3}) \) and our goal is to simplify each term within the parentheses by distributing the \( \sqrt{3} \) and then merging the roots where possible.

Distributive Property

The distributive property is a fundamental algebraic property used in various mathematical operations. It states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b + c) \) is equivalent to \( ab + ac \). This property allows us to distribute a single term across the terms within a parenthesis through multiplication.

In the context of our example, we apply the distributive property through multiplying \( \sqrt{3} \) with both \( 5\sqrt{2} \) and \( \sqrt{3} \) inside the parentheses individually. This results in two separate expressions, \( 5\sqrt{3} \cdot \sqrt{2} \) and \( \sqrt{3} \cdot \sqrt{3} \) respectively, which can then be further simplified.

Square Roots Arithmetic

Arithmetic involving square roots can be tricky, yet it follows specific rules to ensure accurate simplification. When multiplying square roots, you can combine them as follows: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \) if \( a \) and \( b \) are positive numbers.

Another key rule is that the square root of a square number simplifies to the number itself. For example, \( \sqrt{9} \) simplifies to 3 because 9 is a perfect square. This knowledge is particularly useful in the second step of our exercise. When the square root of 3 multiplies by itself, it simplifies to 3 (the number inside), thus changing our expression into \( 5\sqrt{6}+3 \) after combining and simplifying the terms.

• \[\sqrt{3} \cdot 5\sqrt{2} \rightarrow 5\sqrt{3 \times 2} = 5\sqrt{6}\]
• \[\sqrt{3} \cdot \sqrt{3} \rightarrow \sqrt{3^2} = 3\]