Problem 13 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 5 \sqrt{7}+2 \sqrt{7} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(5 \sqrt{7} + 2 \sqrt{7}\) is \(7 \sqrt{7}\)

Step 1: Identify Like Terms

Both terms in this expression, \(5 \sqrt{7}\) and \(2 \sqrt{7}\), are alike because they both have \(\sqrt{7}\) as part. So they can be combined just like normal algebraic like terms.

Step 2: Combine Like Terms

Simplify the expression by combining the like terms. This involves adding the coefficients (the numbers in front of the \(\sqrt{7}\)). Here, add 5 and 2.

Step 3: Write the Final Answer

After adding the coefficients (5 and 2), write the simplified version of the expression by multiplying the sum of the coefficients (7) by \(\sqrt{7}\). So, \(7 \sqrt{7}\) is the final simplified form of the expression.

Key Concepts

Like Terms

When simplifying expressions that contain radicals, it's important to identify like terms. Like terms are terms that have the same variables raised to the same powers. In the case of radicals, terms that have the same radicand (the number under the radical) are considered like terms. So, if you see terms like \(5 \sqrt{7}\) and \(2 \sqrt{7}\), you should recognize that they share the same radicand, \(\sqrt{7}\). This similarity means you can treat these terms as though they belong to the same group, allowing you to combine them in further calculations. This becomes a bit like organizing your schoolwork: items with similar properties or "labels" are placed together, making it easier to work with them.

Coefficients

Coefficients often appear in expressions involving radicals. They are the numerical parts just before a radical or a variable. Think of coefficients as labels that help measure or count the number of radical terms you have. Take the expression \(5 \sqrt{7}\): here, \(5\) is the coefficient that tells us there are five units of \(\sqrt{7}\). Understanding coefficients allows you to easily perform basic operations like addition and subtraction on expressions.

• In the expression \(5 \sqrt{7} + 2 \sqrt{7}\), \(5\) and \(2\) are the coefficients.
• They act as signals that tell us how many times a particular term is being counted in the expression.

Recognizing coefficients simplifies many mathematical processes since you only need to perform operations on these numbers, keeping the radical part consistent. By isolating and handling coefficients, you'll streamline your calculations.

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions, particularly those involving radicals. Once you have identified like terms by noticing they share the same radicand, you're ready to combine them. This means you perform operations like addition or subtraction on their coefficients while the radical part remains unchanged.To illustrate, consider the expression \(5 \sqrt{7} + 2 \sqrt{7}\):

• The like terms here have a common radicand of \(\sqrt{7}\).
• You add their coefficients: \(5 + 2 = 7\).
• The resulting expression becomes \(7 \sqrt{7}\).

Combining like terms helps to neatly and efficiently condense an expression from a more complex form to a simpler one. This step ensures consistency and correctness in mathematics, especially when dealing with larger and more complicated expressions. It’s like tidying up your room — it makes everything look cleaner and more organized!