Problem 42 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ 3(5+\sqrt{21}) $$

Step-by-Step Solution

Answer

The simplified and rationalized form of the expression is \(15 + 3\sqrt{21}\).

Step 1: Understand the given expression

The expression is \(3(5+\sqrt{21})\). This is a multiplication of the integer \(3\) with the sum of \(5\) and \(\sqrt{21}\). Note that \(\sqrt{21}\) is an irrational number, and it is included within the parentheses.

Step 2: Apply the distributive property

The distributive property states that \(a(b + c) = ab + ac\). Apply this to the expression: \(3(5 + \sqrt{21}) = 3 * 5 + 3 * \sqrt{21}\).

Step 3: Simplify the expression

Multiply the numbers to simplify the expression: \(3 * 5 + 3 * \sqrt{21} = 15 + 3\sqrt{21}\).

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle that helps us break down complex expressions. It explains how to multiply a single term by a sum or difference inside parentheses. In simpler terms, when you see something like \(a(b + c)\), you distribute the \(a\) to both \(b\) and \(c\). This means multiplying each term inside the parentheses by the term outside.For example, if we have the expression \(3(5 + \sqrt{21})\), we can distribute the \(3\) as follows:

• Multiply \(3\) by \(5\), which results in \(15\).
• Multiply \(3\) by \(\sqrt{21}\), which gives us \(3\sqrt{21}\).

So, using the distributive property, the expression simplifies to \(15 + 3\sqrt{21}\). This property is valuable because it allows us to simplify complex expressions and makes it easier to work with them.

Irrational Numbers

Irrational numbers are fascinating because they cannot be expressed exactly as a simple fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\). These numbers have non-repeating, non-terminating decimal parts. Examples include \(\sqrt{2}\), \(\pi\), and in our expression, \(\sqrt{21}\).Understanding irrational numbers is crucial in mathematics as they appear in various computations and geometric lengths. While they can't be represented exactly as simple fractions or decimals, they retain their exactness when expressed in radical form, like \(\sqrt{21}\). In the expression \(15 + 3\sqrt{21}\), the \(\sqrt{21}\) remains an irrational number, clearly showcasing how irrational numbers appear in mathematical expressions, even after simplification.

Simplifying Expressions

Simplifying expressions is the process of making them easier to understand and work with. By reducing expressions to their simplest form, you eliminate any unnecessary complications, making the numbers easier to handle.When you simplify an expression like \(3(5 + \sqrt{21})\), you apply mathematical operations to remove brackets and combine like terms. This involves using the distributive property, as described, to rewrite the expression without parentheses, leading to \(15 + 3\sqrt{21}\).Key points to remember when simplifying:

• Apply the distributive property to remove parentheses.
• Combine like terms where possible.
• Express irrational numbers in their simplified radical form.

In this process, keeping expressions as simple as possible ensures accuracy and ease in subsequent calculations and problem solving.