Problem 53 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. $$ 5 \sqrt{7}+3 \sqrt{7} $$

Step-by-Step Solution

Answer

The expression simplifies to \(8 \sqrt{7}\).

Step 1: Identify Like Radicals

To simplify an expression with radicals, first identify terms that have like radicals. In the expression \(5 \sqrt{7} + 3 \sqrt{7}\), both terms have the radical \(\sqrt{7}\). This means they are like radicals.

Step 2: Combine the Coefficients

Since the radicals are like, you can combine them by adding their coefficients. The expression becomes \((5 + 3) \sqrt{7}\).

Step 3: Calculate the Sum

Add the coefficients together: \(5 + 3 = 8\). So the expression simplified by combining like radicals is \(8 \sqrt{7}\).

Key Concepts

Understanding Like Radicals

When working with radical expressions, it's essential to identify 'like radicals.' Like radicals are similar to like terms in algebra, where the variables must match exactly. In radicals, this means that the expressions under the square root (or other roots) should be identical for the radicals to be considered "like."
For instance, in the expression \(5 \sqrt{7} + 3 \sqrt{7}\), both terms contain \(\sqrt{7}\). This makes them like radicals.
Only like radicals can be combined through simple operations such as addition or subtraction, similar to how one would combine like terms in algebra.

Combining Coefficients of Like Radicals

Once like radicals are identified, the next step is to combine their coefficients, much like combining like terms in algebraic expressions. The coefficient is the number directly in front of the radical.
Let's take a closer look at our expression: \(5 \sqrt{7} + 3 \sqrt{7}\). The coefficients here are 5 and 3, which are the numbers in front of the \(\sqrt{7}\).
To combine these, the process is straightforward:

• Add the coefficients together: \(5 + 3\).
• Keep the common radical part \(\sqrt{7}\) as it is.

This yields the simplified expression \(8 \sqrt{7}\), where 8 is the combined coefficient.

Building Algebraic Expressions

Algebraic expressions can often involve terms that contain both numbers and radicals, making it essential to simplify them properly to work with them easily. By simplifying expressions with radicals, you make algebraic expressions neater and more functional.
An expression like \(5 \sqrt{7} + 3 \sqrt{7}\) can look complex at first glance, but by identifying like radicals and combining their coefficients, you simplify the expression to \(8 \sqrt{7}\).
This process allows for more efficient computation in algebra, as simplified expressions are easier to handle in larger mathematical problems.

• Identifying and combining like radicals simplifies your work.
• Think of it as tidying up your algebraic expressions to make them more manageable.

Simplifying helps streamline calculations in various applications.