Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ 8 \sqrt{3}+\sqrt{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(9\sqrt{3}\).

Step 1: Identify the Common Terms

In the given expression, both terms have the square root of 3, \( \sqrt{3} \), as a common term. Our expression is \( 8\sqrt{3} + \sqrt{3} \).

Step 2: Combine Like Terms

Since \( \sqrt{3} \) is a common factor, we can combine the terms. Think of \( 8\sqrt{3} \) as \( 8 \times \sqrt{3} \) and \( \sqrt{3} \) as \( 1 \times \sqrt{3} \). When we combine them, we add the coefficients: \( 8 + 1 = 9 \). Hence, we get \( 9\sqrt{3} \).

Step 3: Write the Expression in Simplest Form

The combined expression from Step 2 is \( 9\sqrt{3} \). This form has no further like terms to combine, so it is in its simplest form.

Key Concepts

Combining Like Terms

When you encounter an expression with similar terms, such as \(8\sqrt{3} + \sqrt{3}\), the process of combining like terms simplifies the expression by adding or subtracting the coefficients of those similar terms. In this case, both terms have \(\sqrt{3}\) as a common factor, which makes them like terms.

• To combine them, focus on their coefficients. Think of the terms as \(8 \times \sqrt{3}\) and \(1 \times \sqrt{3}\).
• Add the numbers in front of the square root (coefficients): \(8 + 1 = 9\).

By combining these like terms, the expression \(8\sqrt{3} + \sqrt{3}\) reduces to \(9\sqrt{3}\). This is because the terms with the same radical part can be treated much like adding regular numbers.

Square Roots

Square roots are a crucial part of simplifying radical expressions. The square root of a number \(a\), written as \(\sqrt{a}\), is the value that, when multiplied by itself, equals \(a\). For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
In expressions like \(8\sqrt{3}\), the number 3 represents the radicand, which is the value under the square root symbol. Key points to consider:

• Square roots can only be combined (added or subtracted) when their radicands are the same, which is why \(8\sqrt{3} + \sqrt{3}\) can be simplified.
• If the radicands are different, such as \(\sqrt{3} + \sqrt{5}\), they cannot be combined using basic arithmetic.

Understanding how to manipulate and simplify square roots is essential for working with radical expressions effectively. This ensures expressions are brought to their simplest form, often defined by having as few radicals as possible.

Coefficients

Coefficients in algebra are the numerical factors in terms, positioned in front of any variables or radical signs they multiply. In the expression \(8\sqrt{3}\), the number 8 is the coefficient of the term. Coefficients play a significant role when simplifying expressions because they dictate how terms with similar radicals can be combined.
Let's break it down:

• Consider the coefficient as a multiplier to the variable part, in this case, \(\sqrt{3}\).
• When combining like terms, only the coefficients are added or subtracted, not the variables or radicals.
• In \(8\sqrt{3} + 1\sqrt{3}\), the coefficients 8 and 1 are combined to make 9, leading to \(9\sqrt{3}\).

It's essential to identify these coefficients correctly to simplify expressions efficiently. They ensure that terms are correctly combined, reflecting the cumulative quantity of each distinct radical or variable segment in an expression.