Problem 19 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify like terms

In this expression, \( \sqrt{3} \) and \( 5 \sqrt{3} \) are like terms because they both contain the exact same radical, which is \( \sqrt{3} \).

Step 2: Combine like terms

Like terms can be combined by adding or subtracting the coefficients (the number in front of the radical). If there is no number in front of the radical, as in \( \sqrt{3} \), it is understood to be 1. So, adding \( \sqrt{3} \) (which is same as \( 1\sqrt{3} \)) and \( 5 \sqrt{3} \), you end up with \( 6\sqrt{3} \).

Key Concepts

Combining Like Terms

When simplifying math expressions, one of the first steps is to combine like terms. Terms are considered 'like' if they have the same variables raised to the same power, including any radicals. For example, in the expression, \

\(\sqrt{3}+5 \sqrt{3}\)

, the two terms are both multiples of \(\sqrt{3}\), making them like terms.

To combine them, simply add or subtract their coefficients, which are the numbers in front of the variables or radicals. If a term doesn't have an explicit coefficient, like \(\sqrt{3}\) in the exercise, it’s assumed to have a coefficient of 1. So \(\sqrt{3}\) is really \(1\sqrt{3}\). When combining \(1\sqrt{3} + 5\sqrt{3}\), you simply add the coefficients (1 + 5) and keep the common radical part \(\sqrt{3}\), resulting in \(6\sqrt{3}\).

For algebraic success, mastering the skill of combining like terms is essential, and it allows for the simplification of many algebraic expressions.

Radical Terms

Radical terms consist of a number or expression under the symbol of a radical, most commonly a square root. The radical sign indicates the root of the number, and in the absence of an index, it is understood to represent the square root. In our exercise, \(\sqrt{3}\) is a radical term that signifies the square root of 3.

Working with radical terms includes the understanding that they can only be combined with other radical terms with the same index and radicand (the number under the radical sign). So \(\sqrt{3}\) can only combine with another \(\sqrt{3}\), but not with \(\sqrt{2}\) or \(\sqrt{3^2}\). Simplifying radical expressions often involves combining these like radicals, or sometimes simplifying them by finding square factors that can be taken out of the root for easier computation.

Simplification of Expressions

The simplification of expressions is a key part of algebra that helps in understanding and solving problems more easily. This process involves reducing an expression to its most basic form without changing its value. It often includes expanding multiplication, combining like terms, factoring, and canceling common factors in fractions.

In the case of radical expressions, simplification may mean identifying and combining like radical terms or rationalizing the denominator. All these steps clear up an expression and pave the way for further solving operations, such as equations or inequalities. Simplified expressions are generally easier to read, and it's easier to spot patterns and solutions when dealing with their simplest form.

Addition of Radicals

The addition of radicals involves combining radical terms that have the same radical component, which is necessary for the terms to be considered like terms. When adding radicals, make sure that both the index (which tells you the degree of the root) and the radicand (the value inside the root) match.

If they do not match, you may need to simplify the radicals first or find a common denominator. For example, \(\sqrt{3}\) and \(5\sqrt{3}\) can be added together because their radicands are the same. On adding, you get \(6\sqrt{3}\), which is the simplified form of the original expression. The process is similar to combining like terms, where you only combine the coefficients and keep the radical part intact.